Solve {l}{text{(a)}(x-y)^2-4(x-y)-45}{text{(a)}(x+3)(x-3)(x^2+9)}

$$\left. \begin{array} { l } { \text { (a) } ( x - y ) ^ { 2 } - 4 ( x - y ) - 45 } \\ { \text { (a) } ( x + 3 ) ( x - 3 ) ( x ^ { 2 } + 9 ) } \end{array} \right.$$

Least Common Multiple

$$a\left(x^{4}-81\right)\left(ax^{2}-2axy-4x+ay^{2}+4y-45\right)$$

Solution Steps

Factor the expressions that are not already factored.

$$ax^{4}-81a=a\left(x-3\right)\left(x+3\right)\left(x^{2}+9\right)$$

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

$$a\left(x-3\right)\left(x+3\right)\left(x^{2}+9\right)\left(ax^{2}-2axy-4x+ay^{2}+4y-45\right)$$

Expand the expression.

$$a^{2}x^{6}-4ax^{5}-45ax^{4}-81a^{2}x^{2}+162xya^{2}+324ax+a^{2}y^{2}x^{4}-81a^{2}y^{2}-2ya^{2}x^{5}+4ayx^{4}-324ay+3645a$$

Evaluate

$$a\left(x-y\right)^{2}+4y-4x-45,\ a\left(x^{4}-81\right)$$