Solve x^{2}y^{2}-a^{2}y^{2}-b^2x^2+a^2b^2

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

Factor

$$\left(x-a\right)\left(y-b\right)\left(x+a\right)\left(y+b\right)$$

Solution Steps

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

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

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

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

Consider x^{2}-a^{2}. The difference of squares can be factored using the rule: p^{2}-q^{2}=\left(p-q\right)\left(p+q\right).

$$\left(x-a\right)\left(x+a\right)$$

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

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

Rewrite the complete factored expression.

$$\left(x-a\right)\left(x+a\right)\left(y-b\right)\left(y+b\right)$$

Evaluate

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