Solve 5u^2v^2-20v^2+15u^2-60

$$5 u ^ { 2 } v ^ { 2 } - 20 v ^ { 2 } + 15 u ^ { 2 } - 60$$

Factor

$$5\left(u-2\right)\left(u+2\right)\left(v^{2}+3\right)$$

Solution Steps

Factor out 5.

$$5\left(u^{2}v^{2}-4v^{2}+3u^{2}-12\right)$$

Consider u^{2}v^{2}-4v^{2}+3u^{2}-12. Do the grouping u^{2}v^{2}-4v^{2}+3u^{2}-12=\left(u^{2}v^{2}-4v^{2}\right)+\left(3u^{2}-12\right), and factor out v^{2} in the first and 3 in the second group.

$$v^{2}\left(u^{2}-4\right)+3\left(u^{2}-4\right)$$

Factor out common term u^{2}-4 by using distributive property.

$$\left(u^{2}-4\right)\left(v^{2}+3\right)$$

Consider u^{2}-4. Rewrite u^{2}-4 as u^{2}-2^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(u-2\right)\left(u+2\right)$$

Rewrite the complete factored expression. Polynomial v^{2}+3 is not factored since it does not have any rational roots.

$$5\left(u-2\right)\left(u+2\right)\left(v^{2}+3\right)$$

Evaluate

$$5\left(u^{2}-4\right)\left(v^{2}+3\right)$$