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

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

Least Common Multiple

$$\left(a+b\right)\left(-x-y+a-b\right)\left(bx-ax-ay-by+\left(a+b\right)^{2}\right)$$

Solution Steps

Factor the expressions that are not already factored.

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

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

$$\left(a+b\right)\left(-x-y+a-b\right)\left(bx-ax-ay-by+2ab+a^{2}+b^{2}\right)$$

Expand the expression.

$$a^{2}x^{2}-b^{2}x^{2}+2abxy+2xya^{2}-2axb^{2}-2bxa^{2}-2xa^{3}-2xb^{3}+a^{2}y^{2}+b^{2}y^{2}+2aby^{2}-4bya^{2}-2ayb^{2}-2ya^{3}+2ba^{3}-2ab^{3}-b^{4}+a^{4}$$

Evaluate

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