Here are some hints:
1) To show $F[x]/(f(x)) \cong F$, it suffices to find a surjective ring homomorphism $\varphi: F[x] \rightarrow F$ whose kernel is $(f(x))$. (That's the first isomorphism theorem at work.) Moreover, assuming you map the constant polynomials to $F$ in the obvious way, the homomorphism $\varphi$ is determined by your choice of the value of $\varphi(x)$. What value of $\varphi(x)$ would result in $f(x) = x - 1$ being in the kernel?
2) The ideal $(g(x))$ is maximal if and only if $F[x] / (g(x))$ is a field. What does the factorization $x^2 - 1 = (x-1)(x+1)$ tell you about the ring $F[x] / (g(x))$? (You can also do this problem directly from the definition of a maximal ideal, but the approach suggested is important to understand.)