Solve {l}{2x^{2}(x^{2}-1)-(2x^{2}-1)(2x^{2}+1)+2}{(2a+b)(b-2a)-(a-3b)(a+3b)}{(x+y)(x-y)(x^{2}+y^2)+x^2(y^2-x^2)}

$$\left. \begin{array} { l } { 2 x ^ { 2 } ( x ^ { 2 } - 1 ) - ( 2 x ^ { 2 } - 1 ) ( 2 x ^ { 2 } + 1 ) + 2 } \\ { ( 2 a + b ) ( b - 2 a ) - ( a - 3 b ) ( a + 3 b ) } \\ { ( x + y ) ( x - y ) ( x ^ { 2 } + y ^ { 2 } ) + x ^ { 2 } ( y ^ { 2 } - x ^ { 2 } ) } \end{array} \right.$$

Least Common Multiple

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

Solution Steps

Factor the expressions that are not already factored.

$$3-2x^{2}-2x^{4}=-2\left(x^{2}-\left(-\frac{1}{2}\sqrt{7}-\frac{1}{2}\right)\right)\left(x^{2}-\left(\frac{1}{2}\sqrt{7}-\frac{1}{2}\right)\right) -5a^{2}+10b^{2}=5\left(-a^{2}+2b^{2}\right) y^{2}x^{2}-y^{4}=\left(x+y\right)\left(x-y\right)y^{2}$$

Identify all the factors and their highest power in all expressions. Multiply the highest powers of these factors to get the least common multiple.

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

Expand the expression.

$$20b^{2}x^{4}y^{4}-10a^{2}x^{4}y^{4}+20b^{2}x^{2}y^{4}-10a^{2}x^{2}y^{4}+30b^{2}x^{2}y^{2}-15a^{2}x^{2}y^{2}+15a^{2}y^{4}-30b^{2}y^{4}+10a^{2}y^{2}x^{6}-20b^{2}y^{2}x^{6}+10a^{2}y^{2}x^{4}-20b^{2}y^{2}x^{4}$$

Evaluate

$$3-2x^{2}-2x^{4},\ 10b^{2}-5a^{2},\ \left(xy\right)^{2}-y^{4}$$