To factor the expression 9x² + 12x + 4, we first look for two numbers that multiply to give us the product of the coefficient of x² (which is 9) and the constant term (which is 4). This product is 36. We also need these two numbers to add up to the coefficient of x (which is 12).
The two numbers that satisfy these conditions are 6 and 6, since 6 * 6 = 36 and 6 + 6 = 12.
Now we can rewrite the middle term (12x) using these numbers:
9x² + 6x + 6x + 4
Next, we group the terms:
(9x² + 6x) + (6x + 4)
Now factor out the common terms in each group:
3x(3x + 2) + 2(3x + 2)
Notice that both groups contain the common factor (3x + 2). We can factor that out:
(3x + 2)(3x + 2)
Thus, the expression in factored form is:
(3x + 2)²