Solve {l}{3n^2(n-3)-(2n+5)^2}{(3a-2b)(3a+2b)(9a^2-4b^2)}

$$\left. \begin{array} { l } { 3 n ^ { 2 } ( n - 3 ) - ( 2 n + 5 ) ^ { 2 } } \\ { ( 3 a - 2 b ) ( 3 a + 2 b ) ( 9 a ^ { 2 } - 4 b ^ { 2 } ) } \end{array} \right.$$

Least Common Multiple

$$\left(3n^{3}-13n^{2}-20n-25\right)\left(9a^{2}-4b^{2}\right)^{2}$$

Solution Steps

Factor the expressions that are not already factored.

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

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

$$\left(3n^{3}-13n^{2}-20n-25\right)\left(9a^{2}-4b^{2}\right)^{2}$$

Expand the expression.

$$-216a^{2}b^{2}n^{3}+48n^{3}b^{4}+243n^{3}a^{4}+936a^{2}b^{2}n^{2}-1053n^{2}a^{4}-208n^{2}b^{4}+1440na^{2}b^{2}-1620na^{4}-320nb^{4}+1800a^{2}b^{2}-2025a^{4}-400b^{4}$$

Evaluate

$$3n^{3}-13n^{2}-20n-25,\ \left(9a^{2}-4b^{2}\right)^{2}$$