Solve {l}{(x+3)(x+5)(x-5)(x-3)}{(a+b)^{2}(a-b)^{2}(a^{4}+a^2b^2+b^4)^2}

$$\left. \begin{array} { l } { ( x + 3 ) ( x + 5 ) ( x - 5 ) ( x - 3 ) } \\ { ( a + b ) ^ { 2 } ( a - b ) ^ { 2 } ( a ^ { 4 } + a ^ { 2 } b ^ { 2 } + b ^ { 4 } ) ^ { 2 } } \end{array} \right.$$

Least Common Multiple

$$\left(x^{2}-25\right)\left(x^{2}-9\right)\left(\left(a^{2}-b^{2}\right)\left(-\left(ab\right)^{2}+\left(a^{2}+b^{2}\right)^{2}\right)\right)^{2}$$

Solution Steps

Factor the expressions that are not already factored.

$$-34x^{2}+x^{4}+225=\left(x-5\right)\left(x-3\right)\left(x+3\right)\left(x+5\right)$$

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

$$\left(x-5\right)\left(x-3\right)\left(x+3\right)\left(x+5\right)\left(a^{2}-b^{2}\right)^{2}\left(a^{2}b^{2}+a^{4}+b^{4}\right)^{2}$$

Expand the expression.

$$x^{4}a^{12}+x^{4}b^{12}-2x^{4}a^{6}b^{6}+68x^{2}a^{6}b^{6}-34x^{2}a^{12}-34x^{2}b^{12}-450a^{6}b^{6}+225a^{12}+225b^{12}$$

Evaluate

$$x^{4}-34x^{2}+225,\ \left(a^{6}-b^{6}\right)^{2}$$