Solve {l}{m^2+2mn+n^2-1}{a^2+4-4a-9n^2}

$$\left. \begin{array} { l } { m ^ { 2 } + 2 m n + n ^ { 2 } - 1 } \\ { a ^ { 2 } + 4 - 4 a - 9 n ^ { 2 } } \end{array} \right.$$

Least Common Multiple

$$\left(\left(2-a\right)^{2}-9n^{2}\right)\left(\left(m+n\right)^{2}-1\right)$$

Solution Steps

Factor the expressions that are not already factored.

$$\left(m+n\right)^{2}-1=\left(m+n-1\right)\left(m+n+1\right) \left(a-2\right)^{2}-9n^{2}=\left(-3n-a+2\right)\left(3n-a+2\right)$$

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

$$\left(2-a-3n\right)\left(m+n-1\right)\left(3n-a+2\right)\left(m+n+1\right)$$

Expand the expression.

$$a^{2}m^{2}-9m^{2}n^{2}-4am^{2}+4m^{2}-18mn^{3}+2mna^{2}-8amn+8mn-9n^{4}+a^{2}n^{2}-4an^{2}+13n^{2}-a^{2}+4a-4$$

Evaluate

$$\left(m+n\right)^{2}-1,\ \left(a-2\right)^{2}-9n^{2}$$