Solve a^2x^4-20ax^2y^2-96{y}^4

$${ a }^{ 2 } { x }^{ 4 } -20a { x }^{ 2 } { y }^{ 2 } -96 { y }^{ 4 }$$

Factor

$$\left(ax^{2}-24y^{2}\right)\left(ax^{2}+4y^{2}\right)$$

Solution Steps

Consider a^{2}x^{4}-20ax^{2}y^{2}-96y^{4} as a polynomial over variable a.

$$x^{4}a^{2}-20x^{2}y^{2}a-96y^{4}$$

Find one factor of the form x^{k}a^{m}+n, where x^{k}a^{m} divides the monomial with the highest power x^{4}a^{2} and n divides the constant factor -96y^{4}. One such factor is ax^{2}-24y^{2}. Factor the polynomial by dividing it by this factor.

$$\left(ax^{2}-24y^{2}\right)\left(ax^{2}+4y^{2}\right)$$

Evaluate

$$\left(ax^{2}-24y^{2}\right)\left(ax^{2}+4y^{2}\right)$$