Dare you to use substitution!

Let $u=\sqrt (x/2) -1$ . Hence $x = 2(u+1)^2$ such that $\frac{dx}{du} = 4(u+1)$ . Now we are integrating $(u+1) u^7 \frac{dx}{du}$ = $(4)(u+1)^2$ $u^7 = 4(u^2 + 2u + 1)u^7 = 4u^9 + 8u^8 + 4u^7$ with respect to u.

Now we have $\frac{2}{5} u^{10}$ + $\frac{8}{9}u^9$ + $\frac{1}{2} u^8$ as our result. Remember to change the limits of 0 and 2 to -1 and 0 coz we are now dealing with u. Your final answer should be - $\frac{1}{90}$ and the result follows.