Expanding Brackets

The method, step by step

1. 1

   Multiply every term inside by the term outside

   3(x + 4) = 3x + 12

   Every term, without exception. Multiplying only the first term is the commonest single-bracket error and it comes from reading the bracket as one object rather than as a sum waiting to be distributed.

2. 2

   Carry the sign with the multiplier

   −2(x − 5) = −2x + 10

   The minus belongs to the 2, so both products are affected. The second term becomes positive because negative times negative is positive. This single line accounts for a large share of all algebra sign errors at GCSE.

3. 3

   For two brackets, form every pair

   (x + 3)(x + 5) = x² + 5x + 3x + 15

   Two terms times two terms gives four products. Working in a fixed order — first with first, first with second, second with first, second with second — means none is missed, which is what mnemonics like FOIL are trying to enforce.

4. 4

   Collect like terms

   x² + 5x + 3x + 15 = x² + 8x + 15

   The two x terms combine; the x² and the constant have nothing to combine with. An expansion is not finished until like terms are gathered, and leaving them separate loses the final mark even though the maths is right.

5. 5

   Check with a number

   Let x = 2: (2+3)(2+5) = 35. And 2² + 8(2) + 15 = 35 ✓

   Substituting any convenient value into both the original and the expanded form must give the same result. This is a complete check that takes fifteen seconds and catches missed terms and sign errors alike. Avoid x = 0 and x = 1, which hide too many mistakes.

The minus sign outside the bracket

More marks are lost to −2(x − 5) than to any other expansion. The minus applies to every term inside, so the answer is −2x + 10, and the second sign changes. Students who multiply the 2 and then attach a minus to the front get −2x − 10, which is wrong by twenty.

The reliable habit is to write the multiplier including its sign, then multiply term by term, deciding each sign as you go. Treating the sign as something to attach at the end is what causes the error.

• Multiply every term inside, not just the first
• The sign belongs to the multiplier
• Two brackets of two terms → four products
• Collect like terms to finish
• Check by substituting a number

Why FOIL runs out

FOIL — first, outer, inner, last — works for exactly one case: two brackets with two terms each. Give a student (x + 2)(x² + 3x + 1) and the mnemonic offers no guidance, because there are six products and no F, O, I or L to point at.

The rule that scales is 'every term in the first bracket multiplies every term in the second'. It handles two by three, three by three, and anything else, so it is the version worth learning even while FOIL still happens to work.

Three brackets, taken two at a time

For (x + 1)(x + 2)(x + 3), expand any two brackets first, then multiply the resulting three-term expression by the remaining bracket. Attempting all three at once produces eight products and a high chance of losing one.

The order does not affect the answer, so it is worth choosing the pair that multiplies most easily. Students who look for that pairing before starting do noticeably less arithmetic.

How we teach expanding

We teach the grid method alongside the linear one. Drawing a box with the terms along the edges makes it structurally impossible to miss a product, and for students who keep dropping terms it converts an error-prone process into a reliable one.

We also require the substitution check on any expansion worth more than one mark. It is the only check that verifies the whole expansion at once, and it is far more reliable than re-reading working the student has just written.