Consider $25x^{4}-34x^{2}y^{2}+9y^{4}$ as a polynomial over variable $x$.
$$25x^{4}-34y^{2}x^{2}+9y^{4}$$
Find one factor of the form $kx^{m}+n$, where $kx^{m}$ divides the monomial with the highest power $25x^{4}$ and $n$ divides the constant factor $9y^{4}$. One such factor is $25x^{2}-9y^{2}$. Factor the polynomial by dividing it by this factor.
Consider $25x^{2}-9y^{2}$. Rewrite $25x^{2}-9y^{2}$ as $\left(5x\right)^{2}-\left(3y\right)^{2}$. The difference of squares can be factored using the rule: $a^{2}-b^{2}=\left(a-b\right)\left(a+b\right)$.
$$\left(5x-3y\right)\left(5x+3y\right)$$
Consider $x^{2}-y^{2}$. The difference of squares can be factored using the rule: $a^{2}-b^{2}=\left(a-b\right)\left(a+b\right)$.
