Question:

Step 1: Apply the Remainder Theorem to find the remainders. When a polynomial is divided by $(x-4)$, the remainder equals the value of the polynomial at $x=4$. Therefore, we need to find $P(4)$ and $Q(4)$. Step 2: Calculate $P(4)$ for the polynomial $P(x) = kx^3 + 3x^2 - 3$. Substituting $x = 4$ into $P(x)$: $$P(4) = k(4)^3 + 3(4)^2 - 3$$ $$P(4) = k(64) + 3(16) - 3$$ $$P(4) = 64k + 48 - 3$$ $$P(4) = 64k + 45$$ Step 3: Calculate $Q(4)$ for the polynomial $Q(x) = 2x^3 - 5x + k$. Substituting $x = 4$ into $Q(x)$: $$Q(4) = 2(4)^3 - 5(4) + k$$ $$Q(4) = 2(64) - 20 + k$$ $$Q(4) = 128 - 20 + k$$ $$Q(4) = 108 + k$$ Step 4: Set the remainders equal and solve for $k$. Since both polynomials leave the same remainder when divided by $(x-4)$, we have: $$P(4) = Q(4)$$ $$64k + 45 = 108 + k$$ Subtracting $k$ from both sides: $$63k + 45 = 108$$ Subtracting $45$ from both sides: $$63k = 63$$ Dividing both sides by $63$: $$k = 1$$ **Final Answer:** The value of $k$ is $\boxed{1}$, which corresponds to **Option 2**.