Multiplying binomials, as area

How do you multiply (x + 2)(x + 3)? The rule everyone learns is FOIL — First, Outer, Inner, Last. It works, but it hides why it works, and it breaks the moment a factor has three terms. The picture underneath is sturdier: a product is an area.

The area model

A rectangle x + 2 wide and x + 3 tall. Slice the width into x and 2, the height into x and 3, and the rectangle falls into four tiles. The whole area is the sum of the tiles:

$(x + 2)(x + 3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6$

FOIL is just the tour of the four tiles — every term of the first factor multiplies every term of the second. That's the actual rule, and it's the distributive property applied twice.

The general quadratic pattern

$(x + a)(x + b) = x^2 + (a + b)x + ab$

The x-coefficient is the sum of the two numbers; the constant is their product. Watch it once more: (x + 5)(x − 2) = x² + 3x − 10 — sum 3, product −10. Burn this pattern in; factoring (next module) is running it backwards.

Beyond two terms

"Every term times every term" needs no rescue when factors grow. (x + 2)(x² + 3x + 1) is a 2-by-3 grid — six tiles:

$x^3 + 3x^2 + x + 2x^2 + 6x + 2 = x^3 + 5x^2 + 7x + 2$

A quick self-check: a 2-term factor times a 3-term factor must produce 2 × 3 = 6 products before combining. Count your terms; if you got five, a tile is missing.

Why this matters

Expanding is the forward direction of the skill that dominates the rest of this course: factoring is the same grid, reconstructed from its area. If you can see the four tiles of (x + a)(x + b), you already know where trinomials come from — and where they'll be taken apart.