Solve 2ax^{2}y^{2}-2a^2y^2-18ax^2+18a^2

$$2 a x ^ { 2 } y ^ { 2 } - 2 a ^ { 2 } y ^ { 2 } - 18 a x ^ { 2 } + 18 a ^ { 2 }$$

Factor

$$2a\left(y-3\right)\left(y+3\right)\left(x^{2}-a\right)$$

Solution Steps

Factor out 2.

$$2\left(ax^{2}y^{2}-a^{2}y^{2}-9ax^{2}+9a^{2}\right)$$

Consider ax^{2}y^{2}-a^{2}y^{2}-9ax^{2}+9a^{2}. Factor out a.

$$a\left(x^{2}y^{2}-ay^{2}-9x^{2}+9a\right)$$

Consider x^{2}y^{2}-ay^{2}-9x^{2}+9a. Do the grouping x^{2}y^{2}-ay^{2}-9x^{2}+9a=\left(x^{2}y^{2}-ay^{2}\right)+\left(-9x^{2}+9a\right), and factor out y^{2} in the first and -9 in the second group.

$$y^{2}\left(x^{2}-a\right)-9\left(x^{2}-a\right)$$

Factor out common term x^{2}-a by using distributive property.

$$\left(x^{2}-a\right)\left(y^{2}-9\right)$$

Consider y^{2}-9. Rewrite y^{2}-9 as y^{2}-3^{2}. The difference of squares can be factored using the rule: p^{2}-q^{2}=\left(p-q\right)\left(p+q\right).

$$\left(y-3\right)\left(y+3\right)$$

Rewrite the complete factored expression.

$$2a\left(x^{2}-a\right)\left(y-3\right)\left(y+3\right)$$

Evaluate

$$2a\left(y^{2}-9\right)\left(x^{2}-a\right)$$