Solve {l}{(a+2b)^{2}-(a-2b)^{2}+(4ab+1)^{2}-(4ab+1)(4ab-1)}{(x+1)^{2}-(x+2)^{2}+(x-1)(x+1)(x^2-1)+(x^2-1)(-x^2+1)}{(a-3b)(-a-3b)+(a-3b)(a+3b)+(a-3b)(3b-a)+a^2+9}

$$\left. \begin{array} { l } { ( a + 2 b ) ^ { 2 } - ( a - 2 b ) ^ { 2 } + ( 4 a b + 1 ) ^ { 2 } - ( 4 a b + 1 ) ( 4 a b - 1 ) } \\ { ( x + 1 ) ^ { 2 } - ( x + 2 ) ^ { 2 } + ( x - 1 ) ( x + 1 ) ( x ^ { 2 } - 1 ) + ( x ^ { 2 } - 1 ) ( - x ^ { 2 } + 1 ) } \\ { ( a - 3 b ) ( - a - 3 b ) + ( a - 3 b ) ( a + 3 b ) + ( a - 3 b ) ( 3 b - a ) + a ^ { 2 } + 9 } \end{array} \right.$$

Least Common Multiple

$$-6\left(2x+3\right)\left(8ab+1\right)\left(2ab-3b^{2}+3\right)$$

Solution Steps

Factor the expressions that are not already factored.

$$16ab+2=2\left(8ab+1\right) 6ab-9b^{2}+9=3\left(2ab-3b^{2}+3\right)$$

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

$$-6\left(2x+3\right)\left(8ab+1\right)\left(2ab-3b^{2}+3\right)$$

Expand the expression.

$$288axb^{3}-192xa^{2}b^{2}+36xb^{2}-312abx-36x+432ab^{3}-288a^{2}b^{2}-468ab+54b^{2}-54$$

Evaluate

$$16ab+2,\ -2x-3,\ 6ab-9b^{2}+9$$