Solve [3x^{2}y^{2}-(-2x^{2}y^{2})+(-1/2x^{2}y^{2})-3/2x^{2}y^2]+x^2+3x^2y^2

$$[ 3 x ^ { 2 } y ^ { 2 } - ( - 2 x ^ { 2 } y ^ { 2 } ) + ( - \frac { 1 } { 2 } x ^ { 2 } y ^ { 2 } ) - \frac { 3 } { 2 } x ^ { 2 } y ^ { 2 } ] + x ^ { 2 } + 3 x ^ { 2 } y ^ { 2 }$$

Factor

$$x^{2}\left(6y^{2}+1\right)$$

Solution Steps

Factor out \frac{1}{2}.

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

Consider 6x^{2}y^{2}+4x^{2}y^{2}-x^{2}y^{2}-3x^{2}y^{2}+2x^{2}+6x^{2}y^{2}. Factor out x^{2}.

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

Consider 6y^{2}+4y^{2}-y^{2}-3y^{2}+2+6y^{2}. Multiply and combine like terms.

$$12y^{2}+2$$

Consider 12y^{2}+2. Factor out 2.

$$2\left(6y^{2}+1\right)$$

Rewrite the complete factored expression.

$$x^{2}\left(6y^{2}+1\right)$$

Simplify. Polynomial 6y^{2}+1 is not factored since it does not have any rational roots.

$$\left(6y^{2}+1\right)x^{2}$$

Evaluate

$$x^{2}+6\left(xy\right)^{2}$$