Question:

**Step 1: Determine the value of $k$.** The given functional equation is $f(x+y) - kxy = f(x) + 2y^2$. Using the given values $f(1)=2$ and $f(2)=8$. Substitute $x=1$ and $y=1$ into the functional equation: $f(1+1) - k(1)(1) = f(1) + 2(1)^2$ $f(2) - k = f(1) + 2$ Substitute the known values of $f(1)$ and $f(2)$: $8 - k = 2 + 2$ $8 - k = 4$ $k = 4$ **Step 2: Rewrite the functional equation.** Substitute $k=4$ into the functional equation: $f(x+y) - 4xy = f(x) + 2y^2$ Rearranging the terms, we get: $f(x+y) = f(x) + 2y^2 + 4xy$ **Step 3: Determine the form of $f(x)$.** Let $f(x) = ax^2 + bx + c$ for some constants $a, b, c$. Then $f(x+y) = a(x+y)^2 + b(x+y) + c = a(x^2 + 2xy + y^2) + bx + by + c = ax^2 + 2axy + ay^2 + bx + by + c$. Substitute $f(x)$ and $f(x+y)$ into the functional equation from Step 2: $ax^2 + 2axy + ay^2 + bx + by + c = (ax^2 + bx + c) + 2y^2 + 4xy$ Subtract $ax^2 + bx + c$ from both sides: $2axy + ay^2 + by = 2y^2 + 4xy$ This equation must hold for all $x, y \in \mathbb{R}$. By comparing the coefficients of $xy$, $y^2$, and $y$: Comparing coefficients of $xy$: $2a = 4 \implies a = 2$. Comparing coefficients of $y^2$: $a = 2$. Comparing coefficients of $y$: $b = 0$. **Step 4: Determine the constant $c$.** With $a=2$ and $b=0$, the function takes the form $f(x) = 2x^2 + c$. Using the given condition $f(1)=2$: $f(1) = 2(1)^2 + c = 2$ $2 + c = 2$ $c = 0$ Thus, the function is $f(x) = 2x^2$. We verify this with the other given condition $f(2)=8$: $f(2) = 2(2)^2 = 2(4) = 8$, which is consistent. **Step 5: Calculate $f(3)$.** Using the derived function $f(x) = 2x^2$: $f(3) = 2(3)^2 = 2(9) = 18$.