Writing elements of field extension in terms of minimal polynomial

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 $\alpha$:

$$ 1 =^{?} (-1 - \alpha^2)(\alpha) = -\alpha -\alpha^3 = -(\alpha + \alpha^3) = -(-1) = 1. $$

.So we have that $\alpha^3 + \alpha + 1 = 0$. Divide by $\alpha$ and transpose, you get $\frac{1}{\alpha} = - \alpha^2 - 1$.

On the other hand, note that $\alpha + 1 = -\alpha^3$, so that $\frac{1}{\alpha+1} = \alpha^{-3} = -(\alpha^2 - 1)^{3}$. Expanding this, and using $\alpha^3 = -\alpha-1$, gives you the answer.

For the first part, we know that $\alpha^3+\alpha+1=0$. What happens when you multiply this by $\alpha^{-1}$?

The second part works in a similar way, starting with $1+\alpha=-\alpha^3$ and using the first part.