power!!!!

This expression is equivalent to:

$\Sigma_{k=0}^{5} C^{5}_{k} [1+(-1)^k](\sqrt{x^3-1})^k x^{5-k}$ .

where $C^{5}_{k} = \frac{5!}{k!(5-k)!}.$ When $k$ is odd, we have cancellation of opposite-sign terms that leaves us with:

$2C^{5}_{0} x^5 + 2C^{5}_{2} (\sqrt{x^3+1})^2 x^3 + 2C^{5}_{4} (\sqrt{x^3+1})^4 x$ ;

or $2C^{5}_{0} x^5 + 2C^{5}_{2} (x^3+1) x^3 + 2C^{5}_{4} (x^3+1)^2 x$ ;

or $2x^5 + 20x^3(x^3+1) + 10x(x^3+1)^2$ ;

or $2x^5 + 20x^6 + 20x^3 + 10x^7 + 20x^4 + 10x$ ;

or $\boxed{10x^7 + 20x^6 + 2x^5 + 20x^4 + 20x^3 + 10x}$

Hence, the polynomial's degree is $\boxed{7}.$