To multiply two binomials like $(x + 3)(x + 2)$, use distribution twice. An area model makes the bookkeeping visible: every term of one factor meets every term of the other.
Por qué funciona
Picture a rectangle with sides $x + 3$ and $x + 2$. Split each side into its two parts and the rectangle splits into four smaller ones: $x \cdot x = x^2$, $x \cdot 2 = 2x$, $3 \cdot x = 3x$ and $3 \cdot 2 = 6$. The whole area is the sum of the parts: $x^2 + 5x + 6$. Algebraically, $(x + 3)(x + 2) = x(x + 2) + 3(x + 2)$, which gives the same four products.
Idea clave
Multiply each term of the first binomial by each term of the second (four products), then combine like terms. The two middle products usually combine.
Worked examples
Ejemplo resuelto: With negatives
Multiply $(x - 4)(x + 7)$.
- Four products: $x \cdot x = x^2$, $x \cdot 7 = 7x$, $-4 \cdot x = -4x$, $-4 \cdot 7 = -28$.
- Combine the middle terms: $7x - 4x = 3x$.
Respuesta: $x^2 + 3x - 28$
Ejemplo resuelto: Leading coefficients
Multiply $(2x + 5)(3x - 1)$.
- $2x \cdot 3x = 6x^2$; $2x \cdot (-1) = -2x$; $5 \cdot 3x = 15x$; $5 \cdot (-1) = -5$.
- Combine: $6x^2 + 13x - 5$.
Respuesta: $6x^2 + 13x - 5$
Ejemplo resuelto: A square of a binomial
Multiply $(x + 3)^2$.
- A square means the factor twice: $(x + 3)(x + 3)$.
- Four products: $x^2$, $3x$, $3x$ and $9$.
- Combine the two middle rectangles: $x^2 + 6x + 9$.
Respuesta: $x^2 + 6x + 9$
Some people remember the four products as FOIL: First, Outer, Inner, Last. It is only a checklist for the area model, and it works only for two binomials; the grid works for any size.
Error común
Writing $(x + 3)^2$ as $x^2 + 9$. Squaring means $(x + 3)(x + 3)$, which has four products, including two middle rectangles of $3x$ each: $x^2 + 6x + 9$. Check with $x = 1$: $(1 + 3)^2 = 16$, but $1 + 9 = 10$.