Final answer:
The quadratic equation 4x^2 + 16x + 12 = 0 can be factored by first taking out a common factor of 4 and then factoring the resulting trinomial to get 4(x + 1)(x + 3) = 0, with solutions x = -1 and x = -3.
Step-by-step explanation:
To factor the quadratic equation 4x^2 + 16x + 12 = 0, we should first look for a common factor in all terms. In this case, all the coefficients are divisible by 4, so we can factor out a 4 to simplify the equation: 4(x^2 + 4x + 3) = 0
Now, we need to factor the quadratic expression in the parentheses. To do this, we look for two numbers that multiply to the constant term, 3, and add up to the linear coefficient, 4. The two numbers that satisfy these conditions are 1 and 3. Therefore, we can write: 4(x + 1)(x + 3) = 0. This is the factored form of the original equation. The solution to the equation is found by setting each factor equal to zero and solving for x: x + 1 = 0 or x + 3 = 0 x = -1 or x = -3. Thus, the solutions are x = -1 and x = -3.