Solve 3/4a^{2}*a^{2}-1/2a^{2}*b^{2}+3b^2*a^2-b^2*2b^2

$$\frac { 3 } { 4 } a ^ { 2 } * a ^ { 2 } - \frac { 1 } { 2 } a ^ { 2 } * b ^ { 2 } + 3 b ^ { 2 } * a ^ { 2 } - b ^ { 2 } * 2 b ^ { 2 }$$

Factor

$$\frac{\left(3a^{2}-2b^{2}\right)\left(a^{2}+4b^{2}\right)}{4}$$

Solution Steps

Factor out \frac{1}{4}.

$$\frac{3a^{2}a^{2}-2a^{2}b^{2}+12b^{2}a^{2}-8b^{2}b^{2}}{4}$$

Consider 3a^{4}-2a^{2}b^{2}+12b^{2}a^{2}-8b^{4}. Multiply and combine like terms.

$$3a^{4}+10a^{2}b^{2}-8b^{4}$$

Consider 3a^{4}+10a^{2}b^{2}-8b^{4}. Consider 3a^{4}+10a^{2}b^{2}-8b^{4} as a polynomial over variable a.

$$3a^{4}+10b^{2}a^{2}-8b^{4}$$

Find one factor of the form ka^{m}+n, where ka^{m} divides the monomial with the highest power 3a^{4} and n divides the constant factor -8b^{4}. One such factor is 3a^{2}-2b^{2}. Factor the polynomial by dividing it by this factor.

$$\left(3a^{2}-2b^{2}\right)\left(a^{2}+4b^{2}\right)$$

Rewrite the complete factored expression.

$$\frac{\left(3a^{2}-2b^{2}\right)\left(a^{2}+4b^{2}\right)}{4}$$

Evaluate

$$\frac{5\left(ab\right)^{2}}{2}+\frac{3a^{4}}{4}-2b^{4}$$