1. **State the problem:** Find the remainder when the polynomial $f(x)$ is divided by $d(x)$ using the Remainder Theorem. 2. **Recall the Remainder Theorem:** If a polynomial $f(x)$ is divided by $x - c$, the remainder is $f(c)$. 3. **Part a:** Given $f(x) = -2x^3 - 3x^2 + 5$ and $d(x) = x - 1$, the divisor is $x - 1$ so $c = 1$. Calculate $f(1)$: $$f(1) = -2(1)^3 - 3(1)^2 + 5 = -2 - 3 + 5 = 0$$ So, the remainder is $0$. 4. **Part b:** Given $f(x) = -3x^{16} + 6x^3 + 75$ and $d(x) = x + 1$, rewrite divisor as $x - (-1)$ so $c = -1$. Calculate $f(-1)$: $$f(-1) = -3(-1)^{16} + 6(-1)^3 + 75 = -3(1) + 6(-1) + 75 = -3 - 6 + 75 = 66$$ So, the remainder is $66$. **Final answers:** - a) Remainder is $0$ - b) Remainder is $66$