1. **State the problem:** Find a polynomial function of degree 4 with zeros at $-2$ and $2$, each with multiplicity 2, and whose graph passes through the point $(-4, 432)$. 2. **Write the general form:** Since the zeros are $-2$ and $2$ with multiplicity 2, the polynomial can be written as: $$f(x) = a(x + 2)^2 (x - 2)^2$$ where $a$ is a constant to be determined. 3. **Use the given point to find $a$:** Substitute $x = -4$ and $f(x) = 432$: $$432 = a(-4 + 2)^2 (-4 - 2)^2$$ Calculate inside the parentheses: $$432 = a(-2)^2 (-6)^2$$ Simplify powers: $$432 = a(4)(36)$$ $$432 = 144a$$ Divide both sides by 144: $$\cancel{144}a = \frac{432}{\cancel{144}}$$ $$a = 3$$ 4. **Write the final polynomial:** $$f(x) = 3(x + 2)^2 (x - 2)^2$$ 5. **Optional expansion:** $$(x + 2)^2 = x^2 + 4x + 4$$ $$(x - 2)^2 = x^2 - 4x + 4$$ Multiply: $$(x^2 + 4x + 4)(x^2 - 4x + 4) = x^4 - 4x^3 + 4x^2 + 4x^3 - 16x^2 + 16x + 4x^2 - 16x + 16$$ Simplify: $$x^4 + ( -4x^3 + 4x^3 ) + (4x^2 - 16x^2 + 4x^2) + (16x - 16x) + 16 = x^4 - 8x^2 + 16$$ Multiply by 3: $$f(x) = 3x^4 - 24x^2 + 48$$ **Final answer in factored form:** $$f(x) = 3(x + 2)^2 (x - 2)^2$$