The Machine in Letters

The rule for factoring can be written with letters. The identity $x^{2}+\left(p+q\right)x+pq = \left(x+p\right)\left(x+q\right)$ holds for any numbers $p$ and $q$. The coefficient of $x$ is their sum, and the constant term is their product.

Expanding $\left(x+p\right)\left(x+q\right)$ gives $x^{2}+\left(p+q\right)x+pq$. Factoring uses the same identity in the other direction. It turns the sum back into the product.

Expand $\left(x+p\right)\left(x+q\right)$ to get $x^{2}+px+qx+pq$. Combine the two $x$ terms to get $x^{2}+\left(p+q\right)x+pq$. This shows why the two numbers must add to the middle coefficient and multiply to the constant.