1. **State the problem:** We are given the function $f(x) = -x(x+2)^2$ and want to understand its behavior and possibly simplify or analyze it. 2. **Recall the formula and rules:** The function is a product of $-x$ and the square of $(x+2)$. We can expand it using the distributive property and the binomial expansion for $(x+2)^2$. 3. **Expand $(x+2)^2$:** $$ (x+2)^2 = x^2 + 4x + 4 $$ 4. **Substitute back into $f(x)$:** $$ f(x) = -x(x^2 + 4x + 4) $$ 5. **Distribute $-x$ across the trinomial:** $$ f(x) = -x^3 - 4x^2 - 4x $$ 6. **Final simplified form:** $$ f(x) = -x^3 - 4x^2 - 4x $$ This is a cubic polynomial with negative leading coefficient, which means the graph will fall to the right and rise to the left. **Answer:** $$ f(x) = -x^3 - 4x^2 - 4x $$