Reading the Product

Last section you expanded (x+2)(x+3) and noticed something in the result x^2 + 5x + 6: the middle coefficient 5 is 2 + 3, and the constant 6 is 2 * 3. The cross products build the middle; the lasts build the end.

That observation, read backwards, is a factoring machine. To factor x^2 + bx + c, find two numbers whose SUM is b and whose PRODUCT is c. Those two numbers drop straight into (x + p)(x + q).

For x^2 + 5x + 6: sum 5, product 6, the pair is 2 and 3, and the factors are (x + 2)(x + 3). One observation, two directions, and the harder direction is now a hunt for a pair of numbers.