To factor the expression x² + 12x - 64, we need to find two binomials that, when multiplied together, give us the original expression.
The factors can be obtained by looking for two numbers whose product is -64 and whose sum is 12.
Let's break down -64 into its factors:
-1 * 64 = -64
-2 * 32 = -64
-4 * 16 = -64
-8 * 8 = -64
Among these pairs, the sum of 8 and -8 gives us 0, so we can rewrite the expression as follows:
x² + 8x - 8x - 64
Now we can group the terms and factor them separately:
x(x + 8) - 8(x + 8)
Now, we can see that we have a common binomial factor, (x + 8), which we can factor out:
(x + 8)(x - 8)
Therefore, the factored form of x² + 12x - 64 is (x + 8)(x - 8).