Solve {l}{(a^{2}-b^{2})(x^2-y^2)+4abxy}{a^2+6a+8-y^2+2y}

$$\left. \begin{array} { l } { ( a ^ { 2 } - b ^ { 2 } ) ( x ^ { 2 } - y ^ { 2 } ) + 4 a b x y } \\ { a ^ { 2 } + 6 a + 8 - y ^ { 2 } + 2 y } \end{array} \right.$$

Least Common Multiple

$$lcm(\left(a^{2}-b^{2}\right)\left(x^{2}-y^{2}\right)+4abxy,\left(4+a-y\right)\left(y+a+2\right))$$

Solution Steps

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

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

Expand the expression.

$$a^{2}b^{2}x^{2}+a^{2}x^{2}y^{2}-b^{2}x^{2}y^{2}-x^{2}a^{4}+6ab^{2}x^{2}-6x^{2}a^{3}+8b^{2}x^{2}-8a^{2}x^{2}+4abxy^{3}-8abxy^{2}-4bxya^{3}-24bxya^{2}-32abxy+b^{2}y^{4}-a^{2}y^{4}+2a^{2}y^{3}-2b^{2}y^{3}+y^{2}a^{4}-a^{2}b^{2}y^{2}-6ab^{2}y^{2}+6y^{2}a^{3}+8a^{2}y^{2}-8b^{2}y^{2}+2yb^{2}x^{2}-2ya^{2}x^{2}$$

Evaluate

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