Solve {l}{(x+y)(2x+5)-(x+y)}{ab^2-bc^2-ab+c^2}{(x-2y)^2+4x-a}

$$\left. \begin{array} { l } { ( x + y ) ( 2 x + 5 ) - ( x + y ) } \\ { a b ^ { 2 } - b c ^ { 2 } - a b + c ^ { 2 } } \\ { ( x - 2 y ) ^ { 2 } + 4 x - a } \end{array} \right.$$

Least Common Multiple

$$2\left(1-b\right)\left(x+2\right)\left(x+y\right)\left(-\left(x-2y\right)^{2}+a-4x\right)\left(ab-c^{2}\right)$$

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.

$$2\left(1-b\right)\left(x+2\right)\left(x+y\right)\left(a-4y^{2}-4x+4xy-x^{2}\right)\left(ab-c^{2}\right)$$

Expand the expression.

$$2ab^{2}x^{4}-2bc^{2}x^{4}+2c^{2}x^{4}-2abx^{4}+12ab^{2}x^{3}-12bc^{2}x^{3}+12c^{2}x^{3}-12abx^{3}+2abc^{2}x^{2}-2a^{2}b^{2}x^{2}+2ba^{2}x^{2}+16ab^{2}x^{2}-16bc^{2}x^{2}-2ac^{2}x^{2}+16c^{2}x^{2}-16abx^{2}+8axb^{2}y^{3}-8bxc^{2}y^{3}+8xc^{2}y^{3}-8abxy^{3}+2abxyc^{2}-2xya^{2}b^{2}+2bxya^{2}+16axyb^{2}-16bxyc^{2}-2axyc^{2}+16xyc^{2}-16abxy+4abxc^{2}-4xa^{2}b^{2}+4bxa^{2}-4axc^{2}+16ab^{2}y^{3}-16bc^{2}y^{3}+16c^{2}y^{3}-16aby^{3}+6byc^{2}x^{3}-6ayb^{2}x^{3}+6abyx^{3}-6yc^{2}x^{3}+4byc^{2}x^{2}-4ayb^{2}x^{2}+4abyx^{2}-4yc^{2}x^{2}+4abyc^{2}-4ya^{2}b^{2}+4bya^{2}-4ayc^{2}$$

Evaluate

$$2\left(x+2\right)\left(x+y\right),\ \left(b-1\right)\left(ab-c^{2}\right),\ \left(x-2y\right)^{2}+4x-a$$