C2.2; E2.2 - Expanding two brackets

Every term times every term

Why does $(x+3)(x+2)$ contain more than $x^2+6$? Picture a rectangle whose side lengths are $x+3$ and $x+2$, for positive $x$. Splitting each side gives four smaller rectangles, not two: their areas are $x^2$, $2x$, $3x$ and $6$.

[DIAGRAM: asset_name: Four areas from two brackets; asset_slug: two_bracket_area; caption: Splitting both sides gives four products, including two middle terms.; description: A rectangle has width x plus three and height x plus two. Vertical and horizontal divisions produce four regions labelled x squared, three x, two x and six. The x-squared region is pale blue, the two middle regions pale sage and the constant region pale clay. The side segments x, three, x and two are labelled outside. The diagram uses positive x; algebraic distribution also handles zero and negative values.]

The same rule follows without a picture:

Each first-bracket term has multiplied the whole second bracket. This algebra works for every real $x$; the area picture is restricted to positive side lengths.

A product grid keeps the four multiplications visible:

Change only $+2$ to $-2$. Two of the four products now contain that negative term.

A grid is an organiser, rather than a different rule. Write negative entries as signed terms in its headers so their signs travel into each product.

Once all four products are present, collect like terms to finish.

Collecting terms after expansion

A number in front of $x$ belongs to its term and must also be multiplied. In $(2x+3)(x+4)$, the first product is $2x\times x=2x^2$; it is not $2x$ and it is not $(2x)^2$.

After multiplying, match the letter parts before collecting. The square terms, the $x$-terms and the constants belong in different groups. An expression with highest power $x^2$ and a nonzero $x^2$-coefficient is called a quadratic expression.

Now change only $+3$ to $-3$. The $x$-coefficient and constant change, but the leading product stays $2x^2$.