To expand a product, multiply every term inside one bracket by every term in the other, keeping each sign with its term, then collect like terms. When a negative term is involved, the skill is in the sign of each product: negative times negative is positive, and negative times positive is negative.
You will need it before factorising with a common factor, because factorising is expanding run backwards.
How do signs behave when you multiply?
Two like signs give a plus, and two different signs give a minus. So −3 × −4 = +12, but −3 × +4 = −12.
Treat the minus as part of the term that follows it. In (2x − 3), the two terms are +2x and −3. That habit is what keeps signs correct.
How to expand, step by step
- Write the terms of each bracket with their signs. For (2x − 3) these are +2x and −3.
- Multiply each term in the first bracket by each term in the second, and write all the products.
- Check the number of products. Two brackets with two terms each give four products.
- Collect like terms, usually the two x terms in the middle.
- Write the answer in descending powers.
Worked example
Expand and simplify (2x − 3)(x − 4)
Step 1, terms: first bracket +2x and −3; second bracket +x and −4.
Step 2, products:
- 2x × x = 2x²
- 2x × (−4) = −8x
- (−3) × x = −3x
- (−3) × (−4) = +12
Step 3, collect: −8x − 3x = −11x.
Step 4, answer:
2x² − 11x + 12
Check: with x = 2, (2x − 3)(x − 4) = (1)(−2) = −2. The answer gives 8 − 22 + 12 = −2. With x = 1, the original gives (−1)(−3) = 3 and the answer gives 2 − 11 + 12 = 3.
The mistake to watch for
A common slip is to apply the negative to the first term inside a bracket only.
Mistaken working: 5 − 2(x − 3) = 5 − 2x − 6 = −1 − 2x
The student multiplied −2 by x correctly, but then wrote −6 instead of +6. They forgot that −2 × −3 is positive.
The correction is to write the products out before combining: −2 × x = −2x and −2 × (−3) = +6. So 5 − 2x + 6 = 11 − 2x.
Substitution shows the error. With x = 1 the original is 5 − 2(−2) = 9. The correct answer gives 11 − 2 = 9, but the mistaken answer gives −1 − 2 = −3.
The same thinking helps with squares of brackets. (x − 3)² is (x − 3)(x − 3), so the four products are x², −3x, −3x and +9, which give x² − 6x + 9.
Check yourself
Try these without a calculator, then open each answer.
1. Expand −4(3a − 2b)
Show answer
−4 × 3a = −12a and −4 × (−2b) = +8b.
−12a + 8b
2. Expand and simplify (x − 5)(x + 2)
Show answer
Products: x², +2x, −5x, −10. The middle terms: 2x − 5x = −3x.
x² − 3x − 10
Check with x = 3: (−2)(5) = −10, and 9 − 9 − 10 = −10.
3. Expand and simplify (3 − x)(2 − x)
Show answer
Products: 3 × 2 = 6, 3 × (−x) = −3x, (−x) × 2 = −2x, (−x) × (−x) = +x². Middle terms: −3x − 2x = −5x.
x² − 5x + 6
Check with x = 4: (−1)(−2) = 2, and 16 − 20 + 6 = 2.
Where this leads next
The previous lesson on collecting terms gives you the final step, and factorising using a common factor runs this process in reverse. Use the non-calculator working trainer to check the number arithmetic inside a product when the coefficients are awkward.
Some students understand the method but still lose a sign under time pressure. A teacher who reads your written working can find that pattern quickly, which is part of online one-to-one Mathematics tuition.