Writing elements of field extension in terms of minimal polynomial 
Let $\alpha \in \mathbb{C}$ be a root of the irreducible polynomial $f(x)=x^3+x+1$. Write $\alpha^{-1}$ and $(\alpha+1)^{-1}$ in terms of $\{1,\alpha,\alpha^2\}$.

So I was able to find a similar question here and was able to solve some of the problems i was given where $\alpha$ had a positive power but I am confused on how to divide the polynomials when it is instead raised to a negative power? Thanks in advance!

Let us assume that $\alpha^{-1} = a \cdot 1 + b \cdot \alpha + c \cdot \alpha^2$ and try and find the coefficients $a,b,c \in \mathbb{C}$. Multiplying both sides of the equation by $\alpha$ we get
$$ 1 = a \cdot \alpha + b \cdot \alpha^2 + c \cdot \alpha^3. $$
Since $\alpha^3 + \alpha + 1 = 0$ we have $\alpha^3 = -\alpha - 1$ and we get
$$ 1 = (-c) \cdot 1 + (a - c) \cdot \alpha + b \cdot \alpha^2 $$
or 
$$ (-c -1) \cdot 1 + (a - c) \cdot \alpha + b \cdot \alpha^2 = 0. $$
Since $1,\alpha,\alpha^2$ are linearly independent, we must have $-c - 1 = a - c = b = 0$ so $a = c = -1$ and $b = 0$ leading to
$$ \alpha^{-1} = -1 - \alpha^2. $$
This can be verified directly by multiplying both sides by $\alpha$:
$$ 1 =^{?} (-1 - \alpha^2)(\alpha) = -\alpha -\alpha^3 = -(\alpha + \alpha^3) = -(-1) = 1. $$
