Solve {l}{8a^{3}+b^3+12a^2b+6ab^2}{27-125a^2-12a+225}

$$\left. \begin{array} { l } { 8 a ^ { 3 } + b ^ { 3 } + 12 a ^ { 2 } b + 6 a b ^ { 2 } } \\ { 27 - 125 a ^ { 2 } - 12 a + 225 } \end{array} \right.$$

Least Common Multiple

$$\left(125a^{2}+12a-252\right)\left(2a+b\right)^{3}$$

Solution Steps

Factor the expressions that are not already factored.

$$252-12a-125a^{2}=-125\left(a-\left(-\frac{12}{125}\sqrt{219}-\frac{6}{125}\right)\right)\left(a-\left(\frac{12}{125}\sqrt{219}-\frac{6}{125}\right)\right)$$

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

$$125\left(a-\frac{-12\sqrt{219}-6}{125}\right)\left(a-\frac{12\sqrt{219}-6}{125}\right)\left(2a+b\right)^{3}$$

Expand the expression.

$$1000a^{5}+96a^{4}-2016a^{3}+125a^{2}b^{3}+72a^{2}b^{2}+12ab^{3}-1512ab^{2}-252b^{3}+750b^{2}a^{3}+1500ba^{4}+144ba^{3}-3024ba^{2}$$

Evaluate

$$\left(2a+b\right)^{3},\ 252-12a-125a^{2}$$