Find a polynomial of degree 3 with real coefficients and zeros of -3, -1, and 4, for which f(-2) = 12.

2(x + 3)(x + 1)(x - 4)

or 2x^3 - 26x - 24.

Step-by-step explanation:

We can write the polynomial if factor form:

P(x) = a(x + 3)(x + 1)(x - 4)  where a is some constant.

Now, since f(-2) = 12  we can write :

12 = a(-2 + 3)(-2 + 1)(-2-4)

12 = 6a

a = 2.

So the polynomial is

2(x + 3)(x + 1)(x - 4).

Expanded that is

2(x + 3)(x^2 - 3x - 4)

= 2(x^3 - 3x^2 - 4x + 3x^2 - 9x - 12)

= 2x^3 - 26x - 24.