Solve ax^{2}+ay^{2}+x^4+2x^2y^2+y^4

$$a x ^ { 2 } + a y ^ { 2 } + x ^ { 4 } + 2 x ^ { 2 } y ^ { 2 } + y ^ { 4 }$$

Factor

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

Solution Steps

Consider ax^{2}+ay^{2}+x^{4}+2x^{2}y^{2}+y^{4} as a polynomial over variable a.

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

Find one factor of the form ka+m, where ka divides the monomial with the highest power \left(x^{2}+y^{2}\right)a and m divides the constant factor x^{4}+2x^{2}y^{2}+y^{4}. One such factor is x^{2}+y^{2}. Factor the polynomial by dividing it by this factor.

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

Evaluate

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