Do the grouping $x^{5}-25-25x^{3}+x^{2}=\left(x^{5}-25x^{3}\right)+\left(x^{2}-25\right)$, and factor out $x^{3}$ in $x^{5}-25x^{3}$.
$$x^{3}\left(x^{2}-25\right)+x^{2}-25$$
Factor out common term $x^{2}-25$ by using distributive property.
$$\left(x^{2}-25\right)\left(x^{3}+1\right)$$
Consider $x^{3}+1$. Rewrite $x^{3}+1$ as $x^{3}+1^{3}$. The sum of cubes can be factored using the rule: $a^{3}+b^{3}=\left(a+b\right)\left(a^{2}-ab+b^{2}\right)$.
$$\left(x+1\right)\left(x^{2}-x+1\right)$$
Consider $x^{2}-25$. Rewrite $x^{2}-25$ as $x^{2}-5^{2}$. The difference of squares can be factored using the rule: $a^{2}-b^{2}=\left(a-b\right)\left(a+b\right)$.
$$\left(x-5\right)\left(x+5\right)$$
Rewrite the complete factored expression. Polynomial $x^{2}-x+1$ is not factored since it does not have any rational roots.
