$$\begin{gathered} \alpha=1^2+4^2+8^2 \ldots . \\ t_n=a^2+b n+c \end{gathered}$$

$$\begin{aligned} & 1=a+b+c \\ & 4=4 a+2 b+c \\ & 8=9 a+3 b+c \end{aligned}$$

On solving we get, $$\mathrm{a}=\frac{1}{2}, \mathrm{~b}=\frac{3}{2}, \mathrm{c}=-1$$

$$\begin{aligned} & \alpha=\sum_{n=1}^{10}\left(\frac{n^2}{2}+\frac{3 n}{2}-1\right)^2 \\ & 4 \alpha=\sum_{n=1}^{10}\left(n^2+3 n-2\right)^2, \beta=\sum_{n=1}^{10} n^4 \\ & 4 \alpha-\beta=\sum_{n=1}^{10}\left(6 n^3+5 n^2-12 n+4\right)=55(353)+40 \end{aligned}$$