Solve x^{4}+x^{2}y^2-2y^4-3x^2-6y^2

$$x ^ { 4 } + x ^ { 2 } y ^ { 2 } - 2 y ^ { 4 } - 3 x ^ { 2 } - 6 y ^ { 2 }$$

Factor

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

Solution Steps

Consider x^{4}+x^{2}y^{2}-2y^{4}-3x^{2}-6y^{2} as a polynomial over variable x.

$$x^{4}+\left(y^{2}-3\right)x^{2}-2y^{4}-6y^{2}$$

Find one factor of the form x^{k}+m, where x^{k} divides the monomial with the highest power x^{4} and m divides the constant factor -2y^{4}-6y^{2}. One such factor is x^{2}+2y^{2}. Factor the polynomial by dividing it by this factor.

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

Evaluate

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