Solve a^4+a^2b^2-156b^4

$$a ^ { 4 } + a ^ { 2 } b ^ { 2 } - 156 b ^ { 4 }$$

Factor

$$\left(a^{2}-12b^{2}\right)\left(a^{2}+13b^{2}\right)$$

Solution Steps

Consider a^{4}+a^{2}b^{2}-156b^{4} as a polynomial over variable a.

$$a^{4}+b^{2}a^{2}-156b^{4}$$

Find one factor of the form a^{k}+m, where a^{k} divides the monomial with the highest power a^{4} and m divides the constant factor -156b^{4}. One such factor is a^{2}+13b^{2}. Factor the polynomial by dividing it by this factor.

$$\left(a^{2}+13b^{2}\right)\left(a^{2}-12b^{2}\right)$$

Evaluate

$$a^{4}-156b^{4}+\left(ab\right)^{2}$$