Multiplying Binomials (FOIL)

To multiply (x + 3)(x + 5), multiply each term in the first bracket by each in the second. FOIL names the four pairs: First, Outer, Inner, Last.

First: x·x = x²
Outer: x·5 = 5x
Inner: 3·x = 3x
Last: 3·5 = 15

Add them and combine the middle: x² + 8x + 15.

Watch the signs. (x − 4)(x + 2) gives x² + 2x − 4x − 8 = x² − 2x − 8.

FOIL only works for two binomials. For anything bigger, use the general rule: every term times every term.

The product is repeated application of the distributive property: (a + b)(c + d) = a(c + d) + b(c + d) = ac + ad + bc + bd. FOIL is a mnemonic for those four products in the 2×2 case and does not extend, which is why it is worth learning the distributive statement instead.

For larger products the box (area) method organises the terms systematically: an m × n grid produces mn products, each combined by degree afterwards. Multiplying a trinomial by a binomial gives six products.

Degrees add under multiplication, so the product of a degree-m and degree-n polynomial has degree m + n, and its leading coefficient is the product of the leading coefficients. That gives a fast check on any expansion: verify the first and last terms before checking the middle.

Reversing this process is factoring, and recognising the structure of the expansion is what makes factoring tractable — the middle coefficient of x² + bx + c is the sum of the two constants, and c is their product.