Solve 4x^{5}y^{3}+(2x^{2}y^{2})(x^{3}y^{3})-(x^{2}y^{3})(-x^{3}y^{2})+2(-4x^{2}y)(x^3y^2)+(4xy^4)(-x^4y)

$$4 x ^ { 5 } y ^ { 3 } + ( 2 x ^ { 2 } y ^ { 2 } ) ( x ^ { 3 } y ^ { 3 } ) - ( x ^ { 2 } y ^ { 3 } ) ( - x ^ { 3 } y ^ { 2 } ) + 2 ( - 4 x ^ { 2 } y ) ( x ^ { 3 } y ^ { 2 } ) + ( 4 x y ^ { 4 } ) ( - x ^ { 4 } y )$$

Factor

$$-\left(y^{2}+4\right)y^{3}x^{5}$$

Solution Steps

Factor out xy.

$$xy\left(4x^{4}y^{2}+2xyx^{3}y^{3}+x^{4}y^{4}-8x^{4}y^{2}-4y^{4}x^{4}\right)$$

Consider 4x^{4}y^{2}+2x^{4}y^{4}+x^{4}y^{4}-8y^{2}x^{4}-4x^{4}y^{4}. Factor out x^{4}y^{2}.

$$x^{4}y^{2}\left(-4-y^{2}\right)$$

Rewrite the complete factored expression. Polynomial -y^{2}-4 is not factored since it does not have any rational roots.

$$xyx^{4}y^{2}\left(-y^{2}-4\right)$$

Evaluate

$$-\left(y^{2}+4\right)y^{3}x^{5}$$