To solve the expression $A^8 - 5A^7 + 7A^6 - 3A^5 + A^4 - 5A^3 + 8A^2 - 2A + I$, we need to work with the matrix $A$ given by:

$A = \begin{bmatrix} 2 & 1 & 1 \\ 0 & 1 & 0 \\ 1 & 1 & 2 \end{bmatrix}$

The calculation involves finding high powers of $A$ up to $A^8$. To simplify this process, we might look for patterns in the powers of $A$, such as identifying any cyclic or null properties (for example, if a certain power of $A$ equals zero, or a power repeats previous results), or we might try diagonalization if applicable.

Here's the step-by-step outline of the solution process:

1. Calculate Initial Powers of $A$: Begin with $A^2, A^3,$ and so forth, checking for any repetitive or simplifying patterns.

2. Identify Patterns or Use Minimal Polynomial: If we find a minimal polynomial of $A$, this could reduce the computational workload by allowing us to express higher powers in terms of lower ones.

3. Combine Terms: Using the powers calculated, substitute back into the expression to find the resulting matrix.

Let’s proceed with calculating these powers explicitly and examining possible simplifications. I'll begin the calculations.The powers of $A$ up to $A^8$ are as follows: