Solve {l}{2x^{2}+3x-6}{2x^2+3x-6}{4x^2+2x+6}{2x^2+3x+5}{4x^2-10x+2}

$$\left. \begin{array} { l } { 2 x ^ { 2 } + 3 x - 6 } \\ { 2 x ^ { 2 } + 3 x - 6 } \\ { 4 x ^ { 2 } + 2 x + 6 } \\ { 2 x ^ { 2 } + 3 x + 5 } \\ { 4 x ^ { 2 } - 10 x + 2 } \end{array} \right.$$

Least Common Multiple

$$\frac{\left(2x^{2}+x+3\right)\left(2x^{2}+3x+5\right)\left(\left(4x-5\right)^{2}-17\right)\left(\left(4x+3\right)^{2}-57\right)}{32}$$

Solution Steps

Factor the expressions that are not already factored.

$$2x^{2}+3x-6=2\left(x-\left(-\frac{1}{4}\sqrt{57}-\frac{3}{4}\right)\right)\left(x-\left(\frac{1}{4}\sqrt{57}-\frac{3}{4}\right)\right) 2x^{2}+3x-6=2\left(x-\left(-\frac{1}{4}\sqrt{57}-\frac{3}{4}\right)\right)\left(x-\left(\frac{1}{4}\sqrt{57}-\frac{3}{4}\right)\right) 4x^{2}+2x+6=2\left(2x^{2}+x+3\right) 4x^{2}-10x+2=2^{2}\left(x-\left(-\frac{1}{4}\sqrt{17}+\frac{5}{4}\right)\right)\left(x-\left(\frac{1}{4}\sqrt{17}+\frac{5}{4}\right)\right)$$

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

$$8\left(x-\frac{-\sqrt{57}-3}{4}\right)\left(x-\frac{5-\sqrt{17}}{4}\right)\left(x-\frac{\sqrt{57}-3}{4}\right)\left(x-\frac{\sqrt{17}+5}{4}\right)\left(2x^{2}+x+3\right)\left(2x^{2}+3x+5\right)$$

Expand the expression.

$$32x^{8}+32x^{7}-112x^{6}-176x^{5}-462x^{4}+338x^{3}-54x^{2}+822x-180$$

Evaluate

$$2x^{2}+3x-6,\ 2x^{2}+3x-6,\ 4x^{2}+2x+6,\ 2x^{2}+3x+5,\ 4x^{2}-10x+2$$