Expanding and factoring

Expanding brackets and collecting like terms, the special products — mental arithmetic included — and factoring.

How to solve it

Expanding brackets and factoring are the same move in opposite directions. The special products work both ways: $(a - b)(a + b) = a^2 - b^2$ expands brackets left to right and factors a difference of squares right to left. They also help with mental arithmetic: $68 \cdot 52 = (60 + 8)(60 - 8) = 3600 - 64$.

Step by step

Expanding: multiply every term of the first bracket by every term of the second, minding the signs.
Collect like terms: add the coefficients of equal powers of $x$.
Recognize a formula when there is one: the square of a sum or a difference, the difference of squares.
Factoring: take out the common factor first (both the number and the letter), then look at what is left in the brackets — it may be a formula.

The first term.

The second term.

Twice the product, the term most often forgotten.

Example: $x^3 - x = x(x^2 - 1) = x(x - 1)(x + 1)$: the common factor first, then the difference of squares.

Common mistakes

Losing twice the product: $(x + 3)^2 = x^2 + 6x + 9$, not $x^2 + 9$.
The sign before a bracket: $x(x - 3) - 2(x + 1) = x^2 - 3x - 2x - 2$ — the minus changes the sign of both terms of the second bracket.
Not taking out the whole common factor: $20x^2 - 15x = 5x(4x - 3)$, not $5(4x^2 - 3x)$.
Stopping too early: $x(x^2 - 1)$ factors further.