So we divide $x^5 + \alpha^{a+1}x^4 + \alpha^{2(a+1)}x^3 + \alpha^{3(a+1)}x^2 + \alpha^{4(a+1)}x + \alpha^{5(a+1)}$ by $x- \alpha^{a+2}$ , right?

I got that $$ x^5 + \alpha^{a+1}x^4 + \alpha^{2(a+1)}x^3 + \alpha^{3(a+1)}x^2 + \alpha^{4(a+1)}x + \alpha^{5(a+1)}=(x- \alpha^{a+2}) (x^4+ \alpha^{a+1}(\alpha+1) x^3+ \alpha^{2a+2}[\alpha^2+ \alpha-1] x^2+ \alpha^{3a+3}[1+ \alpha [\alpha^2+ \alpha-1]]x+ \alpha^{4a+4}[1+ \alpha(1+ \alpha(\alpha^2+\alpha-1) )])+ \alpha^{5a+5}[\alpha [1+ \alpha (1+ \alpha(\alpha^2+ \alpha+1))]]$$

Is it again wrong?

Ok, I will retry them...