What is the key rule for expanding brackets homework?

Multiply the complete outside factor, including its sign, by every term inside the bracket. Collect like terms only after the products are written correctly.

What does a minus before a bracket mean?

It means multiplying the whole bracket by −1. For example, −(3a − 5) = −3a + 5. Do not apply the sign change twice.

Does checking one value prove an expansion?

No. A mismatch disproves the proposed equivalence at that value, but matching values do not prove an identity. Justify the algebraic steps as well.

Can Lirno finish this homework for me?

Use it only within school rules to seek a focused hint or explanation after your attempt. Verify the recognised symbols and reasoning; AI can make mistakes and offers no correctness or grade guarantee.

Expanding brackets homework: negative signs

Expanding brackets homework can go wrong even when you know the multiplication tables. A factor reaches only the first term, a minus sign disappears, or unlike terms get combined. The remedy is to show which signed factor multiplies each term before you simplify anything. A neat final answer is less useful than a line of working you can inspect.

This guide tackles one assignment: expand and simplify 4(2x − 3) − 2(x + 5). All examples below are original practice tasks. You will make your own attempt, inspect the complete expression and use a Lirno hint or explanation only where your school permits it. The images illustrate studying and contain no answer keys. Keep the teacher’s required notation and amount of working when preparing your final submission.

1. Read expanding brackets homework as multiplication

In 4(2x − 3), the 4 multiplies the whole bracket. The distributive property rewrites that product as 4 × 2x + 4 × (−3), giving 8x − 12. Thinking of the terms as signed pieces makes the subtraction easier to track. You are preserving the value of the expression for every allowed value of x, not choosing a value for x.

Compare 4(2x − 3) with 4 × 2x − 3. The second expression multiplies only 2x by 4; it is not another way of writing the first. At x = 0 the first gives −12 and the second gives −3. That simple comparison identifies a missing multiplication without needing to solve an equation or read a long explanation.

RMIT’s introduction to expansion provides a reference for the distributive law. Use it to confirm the rule, then work through the original assignment here. Expanding is a change of form. Unless the question gives an equation to solve, do not invent an equals-zero condition or report a single value of the variable as the answer.

2. Capture the full expression before asking for help

Write the assignment on paper and try the first expansion before opening an explanation. Include the instruction “expand and simplify”, both brackets, every sign and your current working. A cropped photograph that omits the factor before a bracket can turn a correct explanation of the recognised text into an incorrect response to your actual homework.

If school rules allow Lirno, use the homework capture workflow to include the complete task. Check the recognised expression character by character against the paper: 4, the first bracket, 2x, minus 3, then minus 2 multiplying the second bracket. Make sure x has not become a multiplication symbol and a small minus has not become a dash or disappeared.

Do not proceed with a wrong transcription. Correct it or write the expression clearly before asking anything mathematical. Keep names, grades and classmates’ details outside the image, and do not upload restricted assessment material. If photographing the worksheet is not permitted, continue with your paper attempt and ask the teacher directly. Tool access never supplies permission that your course has not given.

A student photographs an entire blank worksheet at a library desk

3. Keep a negative factor attached to its sign

The second part of the assignment is −2(x + 5). Treat the complete outside factor as −2. Its two products are (−2) × x and (−2) × 5, so this part becomes −2x − 10. A common incorrect line is −2x + 10: the second multiplication has lost the sign of its factor.

A minus sign by itself before a bracket is an invisible factor of −1. Thus −(3a − 5) becomes −3a + 5, because (−1) × (−5) is positive. Do not multiply by −1 and then change every sign a second time. That would apply the operation twice and return a different expression.

Write one intermediate line when signs feel uncertain. For −3(2y − 4), show (−3) × 2y + (−3) × (−4), then −6y + 12. At y = 1, the original is −3 × (−2) = 6 and the result is also 6. The intermediate line explains where the positive constant comes from; it is not a decorative extra step.

Unmarked paper strips and wooden tiles lie beside a blank notebook on a rose tabletop

4. Expand first, then collect like terms

For the full task, first write 4(2x − 3) − 2(x + 5) = 8x − 12 − 2x − 10. Only after both products are complete should you combine terms. The variable terms are 8x and −2x; their sum is 6x. The constants are −12 and −10; their sum is −22. The simplified expression is 6x − 22.

The terms 6x and −22 cannot be merged into −16x or any other single term of that kind. They have different variable parts. To see why, compare x = 0 and x = 1: 6x − 22 changes from −22 to −16, whereas −16x changes from 0 to −16. Agreement at x = 1 alone would conceal the error.

Keep each sign with the term as you reorganise a line. If you rewrite 8x − 12 − 2x − 10 as 8x − 2x − 12 − 10, you have moved the signed terms without changing them. Do not turn a subtraction into addition merely because you moved it across the page. There is no equation boundary being crossed in this collection step.

5. Ask for a hint about the first uncertain step

Instead of requesting the entire answer, ask a focused question such as: “In my line 8x − 12 − 2x + 10, which product should I check? Give one hint without finishing the simplification.” That request keeps your attempt visible and narrows the explanation to the sign error. You still need to judge whether the response matches the original expression.

The Lirno Tutor explanation workflow can support this conversation where permitted. Read the proposed hint, identify the factor and term it refers to, and rewrite the line yourself. If the explanation skips directly to a final answer, ask why the product of −2 and +5 has the sign it does. Compare the reasoning with your class rule.

AI can misread symbols and reason incorrectly. A fluent answer is not a mathematical check, and Lirno does not guarantee correctness, grades, mastery or permission. It is free to download or start, while some AI usage levels or advanced features may require Premium. Use help to understand a transformation you can explain, not to conceal authorship or bypass a rule about independent homework.

6. Check by substitution without mistaking it for proof

Test the original and your simplified form separately. At x = 0, the original is 4(−3) − 2(5) = −12 − 10 = −22. The result 6x − 22 gives −22 as well. At x = 3, the original is 4(3) − 2(8) = 12 − 16 = −4, and 6 × 3 − 22 also gives −4.

A mismatch proves the proposed forms are not equivalent for that tested value. Agreement at one or two values does not prove equivalence for every value. The proof here is the sequence of valid distributive and collection steps. Use substitution as an error detector alongside that reasoning, especially with zero, a negative value or a value that makes a bracket vanish.

Do not substitute into your own mistaken transcription twice and call that an independent check. Read the original directly from the assignment and the result from your working. If a check fails, compare each expanded product before redoing the whole page. The guide to finding the first incorrect step describes that narrower way of reviewing a solution.

A student compares two blank pages on a courtyard bench

7. Handle nested brackets and variable factors carefully

A nested expression needs its grouping preserved. In 2[3x − (x − 4)], deal with the inner subtraction first: 3x − x + 4 = 2x + 4. Then multiply the outer bracket by 2 to obtain 4x + 8. Expanding the outer factor first can also work, but only if every remaining group and sign is retained correctly.

A variable factor is still multiplied by each term. For 2x(x − 3), the products are 2x × x = 2x² and 2x × (−3) = −6x. The result is 2x² − 6x, not 2x − 6. These terms are unlike because the powers of x differ. Do not add their coefficients as though both terms were multiples of x.

Two brackets multiplied together are a further task: every term of the first multiplies every term of the second. That is different from adding two brackets or subtracting one bracket. Before using a memorised pattern, name the operation connecting the groups. If your homework covers only a single outside factor, finish that skill rather than guessing a shortcut for a new structure.

8. Finish with a fresh attempt and readable working

Close the explanation and try 3(2x − 5) − 4(x − 2). The expanded line is 6x − 15 − 4x + 8, giving 2x − 7. Check at x = 0: both forms give −7. Next try −(3a − 5) + 2(a + 1): expanding gives −3a + 5 + 2a + 2, then −a + 7. Explain each positive constant before looking back.

If either task fails, record the specific decision that needs attention: distributing to the second term, carrying the negative factor, or collecting constants. A useful practice question changes that feature while keeping the rest manageable. Copying ten completed answers does not show whether you can now make the decision independently. Keep the original failed line available so the correction remains understandable.

For submission, retain the original expression, one complete expanded line, the collected result and any check your teacher requests. Use equality signs only between expressions that really have the same value. The finished page should make your reasoning visible to another reader. Ask for clarification when a symbol or instruction remains uncertain, and follow your school’s requirements for declaring any assistance.

Expanding brackets homework: keep every sign