Solving systems of equations with Modular Arithmetic can be complex, especially with the following equations:
$$(a_0x+{a_0}^2)^e \equiv C_0\;(mod\;n)$$ $$(a_1x+{a_1}^2)^e \equiv C_1\;(mod\;n)$$
My problem is to solve for $x$ (I am given the values of everything else). I have attempted to solve these equations to the best of my ability, but I am not even sure how I would begin to simplify this. I have been told that the interesting $am+a^2$ relates the two $C$'s in a helpful manner, but that is all.
Powers often simplify modulos. For example, it is well known that all cubes are either $0$, $1$, or $-1$ $($mod $7)$. You can verify this yourself by simply cubing all numbers modulo $7$ (that is, cube $0,1,2,3,4,5,$ and $6$ and you will find all results are $0$, $1$, or $-1$ $($mod $7)$ ). Here you have two numbers raised to a power $e$. Without too much difficulty you should be able to find the set of all modulo values $($mod $n)$ that result from raising a number to the $e$th power. $C_0$ and $C_1$ will be two of those. Match them to the corresponding modulo you raised to the $e$th power to get the set of possible values for $a_0x+a_0^2$ and $a_1x+a_1^2$ $($mod $n)$. From there it is merely a matter of using some simple algebra to arrive at your solution.