$$\begin{aligned} & \alpha^6+\alpha^4+\beta^4-5 \alpha^2 \\ & =\alpha^4(\alpha-2)+\alpha^4-5 \alpha^2+(\beta-2)^2 \\ & =\alpha^5-\alpha^4-5 \alpha^2+\beta^2-4 \beta+4 \\ & =\alpha^3(\alpha-2)-\alpha^4-5 \alpha^2+\beta-2-4 \beta+4 \\ & =-2 \alpha^3-5 \alpha^2-3 \beta+2 \\ & =-2 \alpha(\alpha-2)-5 \alpha^2-3 \beta+2 \\ & =-7 \alpha^2+4 \alpha-3 \beta+2 \\ & =-7(\alpha-2)+4 \alpha-3 \beta+2 \\ & =-3 \alpha-3 \beta+16=-3(1)+16=13 \end{aligned}$$