If $alpha$ and $beta$ are the roots of $x^2-4x+5=0$, find $(alpha^3-4alpha^2+6alpha+2)(beta^3-4beta^2+6beta+2)$

$$ left( x^{3} - 4 x^{2} + 6 x + 2 right) = left( x^{2} - 4 x + 5 right) cdot left( x right) + left( x + 2 right) $$

This means that your product is just $$ (alpha+2)(beta+2) = (alpha beta) + 2(alpha + beta) + 4 $$

which you should now be able to evaluate