1. **Problem statement:** We have a polynomial $f(x)$ divisible by $x+2$, and when divided by $x^2$, the remainder is $-20x + 8$. We want to find $R(-1)$ where $R(x)$ is the remainder when $f(x)$ is divided by $x^3 + 2x^2$. 2. **Key facts and formulas:** - If $f(x)$ is divisible by $x+2$, then $f(-2) = 0$. - When dividing by $x^2$, the remainder is a polynomial of degree less than 2, here $-20x + 8$. - When dividing by $x^3 + 2x^2 = x^2(x+2)$, the remainder $R(x)$ must have degree less than 3. 3. **Express $R(x)$:** Since $x^3 + 2x^2 = x^2(x+2)$, the remainder $R(x)$ when dividing by this cubic can be written as $$R(x) = A x^2 + B x + C$$ for some constants $A, B, C$. 4. **Using the division properties:** Since $f(x) = Q(x)(x^3 + 2x^2) + R(x)$, substitute $x = -2$: $$f(-2) = Q(-2) \cdot 0 + R(-2) = R(-2)$$ But $f(-2) = 0$ (divisible by $x+2$), so $$R(-2) = 0$$ 5. **Using the remainder when dividing by $x^2$:** When dividing $f(x)$ by $x^2$, remainder is $-20x + 8$, so $$f(x) = S(x) x^2 + (-20x + 8)$$ Substitute this into the expression for $f(x)$: $$Q(x)(x^3 + 2x^2) + R(x) = S(x) x^2 + (-20x + 8)$$ Rearranged: $$R(x) - (-20x + 8) = S(x) x^2 - Q(x)(x^3 + 2x^2)$$ The right side is divisible by $x^2$, so the left side must be divisible by $x^2$. 6. **Check divisibility by $x^2$ of $R(x) + 20x - 8$:** $$R(x) + 20x - 8 = A x^2 + B x + C + 20x - 8 = A x^2 + (B + 20) x + (C - 8)$$ For this to be divisible by $x^2$, the coefficients of $x$ and constant term must be zero: $$B + 20 = 0 \Rightarrow B = -20$$ $$C - 8 = 0 \Rightarrow C = 8$$ 7. **Use $R(-2) = 0$ to find $A$:** $$R(-2) = A (-2)^2 + B (-2) + C = 4A - 2B + C = 0$$ Substitute $B = -20$, $C = 8$: $$4A - 2(-20) + 8 = 0 \Rightarrow 4A + 40 + 8 = 0 \Rightarrow 4A + 48 = 0 \Rightarrow 4A = -48 \Rightarrow A = -12$$ 8. **Final remainder polynomial:** $$R(x) = -12 x^2 - 20 x + 8$$ 9. **Find $R(-1)$:** $$R(-1) = -12 (-1)^2 - 20 (-1) + 8 = -12 + 20 + 8 = 16$$ **Answer:** $R(-1) = 16$ which corresponds to option C.