algebra, a root of polynomial

Find $\displaystyle \lambda, \mu, v \in \{0,1\}

$ so that $\displaystyle (\alpha^2 + \alpha + 1)(\alpha^2 + 1) = \lambda\alpha^2 + \mu\alpha + v $

Solution from lecturer:

$\displaystyle (\alpha^2 + \alpha + 1)(\alpha^2 + 1) = \alpha^4 + \alpha^3 + \alpha+ 1 = \alpha^4 = \alpha^2 + \alpha $

Therefore $\displaystyle \lambda = \mu = 1, v = 0 $

I'm having real trouble understanding this solution...

I'm guessing this bit $\displaystyle \alpha^4 + \alpha^3 + \alpha+ 1 $ comes from multiplying out and then removing any repeated elements? But why is this then equal to $\displaystyle \alpha^4 = \alpha^2 + \alpha$?

Please could someone please explain this simply?

Thanks in advance!