Deciding which statements about matrix A is true where $A^3+A^2-3A+I=0$

Your answers for (c) and (d) look good (though I admit I haven't crunched the algebra for your answer to part (d)).

For parts (a) and (b), you have a polynomial $p(x)$ such that $p(A) = 0$. The eigenvalues of $A$ must be roots of $p(x)$, but (importantly) not vice-versa. There are more roots than dimensions, meaning that at least one root is not an eigenvalue.

That said, we now have a list of possible eigenvalues that we can have:

1. $\lambda_1 = \lambda_2 = 1$
2. $\lambda_1 = 1$, $\lambda_2 = -1 + \sqrt{2}$
3. $\lambda_1 = 1$, $\lambda_2 = -1 - \sqrt{2}$
4. $\lambda_1 = -1 + \sqrt{2}$, $\lambda_2 = -1 + \sqrt{2}$
5. $\lambda_1 = -1 + \sqrt{2}$, $\lambda_2 = -1 - \sqrt{2}$
6. $\lambda_1 = -1 - \sqrt{2}$, $\lambda_2 = -1 - \sqrt{2}$

All 6 possibilities above are possible. In fact, put each of these pair of numbers into a diagonal matrix, and you'll have a matrix $A$ that satisfies the given polynomial equation, with the given pair of eigenvalues. Note that, in cases 4, 5, 6, $1$ is not an eigenvalue, so (a) is false. Also, in all cases except 1, the determinant is not $1$, so (b) is false.