1. **Stating the problem:** We are given the polynomial $$P(x) = 3x(x - 2)(x + 2)$$ and asked to analyze its graph shape. 2. **Formula and roots:** The polynomial is already factored. The roots are the values of $x$ that make $P(x) = 0$: $$3x(x - 2)(x + 2) = 0 \implies x = 0, 2, -2$$ 3. **Degree and leading term:** Expanding the factors inside: $$P(x) = 3x(x^2 - 4) = 3x^3 - 12x$$ This is a cubic polynomial with leading term $3x^3$. 4. **Behavior of cubic polynomials:** Cubic polynomials with positive leading coefficient behave like $x^3$ for large $|x|$: - As $x \to -\infty$, $P(x) \to -\infty$ - As $x \to +\infty$, $P(x) \to +\infty$ 5. **Graph shape:** The polynomial has three real roots at $-2, 0, 2$ and is cubic-shaped. It passes through the origin and crosses the x-axis at these points. 6. **Matching with given graphs:** Among the described graphs: - Graph A: cubic-like curve with roots at $-2, 0, 2$ and shape consistent with $P(x)$. - Graph D: cubic-shaped curve with roots at $-2, 0, 2$ but described as increasing steeply from negative to positive y-values. Since $P(x)$ has leading term $3x^3$ (positive), it decreases to negative infinity on the left and increases to positive infinity on the right, matching the description of Graph D. **Final answer:** The graph of $$P(x) = 3x(x - 2)(x + 2)$$ corresponds to graph D.