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

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

Least Common Multiple

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

Solution Steps

Factor the expressions that are not already factored.

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

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

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

Expand the expression.

$$-6a^{2}b^{2}x^{2}+3x^{2}a^{4}+3x^{2}b^{4}-3b^{2}x^{2}+10xya^{2}b^{2}-5xya^{4}-5xyb^{4}+5xyb^{2}+y^{2}a^{4}+y^{2}b^{4}-2a^{2}b^{2}y^{2}-b^{2}y^{2}+ya^{4}+yb^{4}-2ya^{2}b^{2}-yb^{2}$$

Evaluate

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