Find the remainder when $(5x + 9)^{611} + (x + 5)^{11} + (x - 1)^{11} + 3x^2 + 1$ is divided by $x + 2.$
This looks like a problem that can be solved using the remainder theorem, which says that the remainder when a polynomial $f(x)$ is divided by $x - a$ is equal to $f(a).$ In this case, we want to find the remainder when the given polynomial is divided by $x + 2$, so we can set $x - a = x + 2$ and solve for $a.$ Subtracting $x$ from both sides, we get $-a = 2$, so $a = -2.$ Now, to find the remainder, we just need to plug in $-2$ for $x$ in the given polynomial and simplify. We get $(5(-2) + 9)^{611} + ((-2) + 5)^{11} + ((-2) - 1)^{11} + 3(-2)^2 + 1 = (-1)^{611} + 3^{11} + (-3)^{11} + 12 + 1.$ Using the fact that $(-1)^{611} = -1$ and that $3^{11} + (-3)^{11} = 0$ (since they are the same magnitude but opposite signs), we can simplify further to get $-1 + 0 + 12 + 1 = 12.$ Therefore, the remainder is 12. 