Consider $2a^{2}x^{2}-a^{3}-2x^{2}a^{2}+1$. Multiply and combine like terms.
$$-a^{3}+1$$
Consider $-a^{3}+1$. By Rational Root Theorem, all rational roots of a polynomial are in the form $\frac{p}{q}$, where $p$ divides the constant term $1$ and $q$ divides the leading coefficient $-1$. One such root is $1$. Factor the polynomial by dividing it by $a-1$.
$$\left(a-1\right)\left(-a^{2}-a-1\right)$$
Rewrite the complete factored expression. Polynomial $-a^{2}-a-1$ is not factored since it does not have any rational roots.
