The problem asks to find the remainder when the polynomial $f(x) = x^4 + 3x^3 - 4x^2 + 3x + 6$ is divided by $(x-2)$.

1. Problem Description

The problem asks to find the remainder when the polynomial $f(x) = x^4 + 3x^3 - 4x^2 + 3x + 6$ is divided by $(x-2)$.

2. Solution Steps

We can use the Remainder Theorem, which states that if we divide a polynomial $f(x)$ by $(x-c)$, then the remainder is $f(c)$. In this case, we are dividing by $(x-2)$, so $c=2$. We need to find $f(2)$.

$f(x) = x^4 + 3x^3 - 4x^2 + 3x + 6$

$f(2) = (2)^4 + 3(2)^3 - 4(2)^2 + 3(2) + 6$

$f(2) = 16 + 3(8) - 4(4) + 6 + 6$

$f(2) = 16 + 24 - 16 + 6 + 6$

$f(2) = 40 - 16 + 12$

$f(2) = 24 + 12$

$f(2) = 36$

3. Final Answer

The remainder is 36.