Solve {l}{5b(-b^2-2b+5)}{2x^4+x^3+2x^2}

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

Least Common Multiple

$$5bx^{2}\left(4bx^{2}-10x^{2}+xb^{2}+2bx-5x+2\left(bx\right)^{2}+2b^{2}+4b-10\right)$$

Solution Steps

Factor the expressions that are not already factored.

$$5b\left(-b^{2}-2b+5\right)=-5b\left(b-\left(\sqrt{6}-1\right)\right)\left(b-\left(-\sqrt{6}-1\right)\right)$$

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

$$5bx^{2}\left(2b^{2}x^{2}+4bx^{2}-10x^{2}+xb^{2}+2bx-5x+2b^{2}+4b-10\right)$$

Expand the expression.

$$10b^{3}x^{4}+20b^{2}x^{4}-50bx^{4}+5b^{3}x^{3}+10b^{2}x^{3}-25bx^{3}+10x^{2}b^{3}+20b^{2}x^{2}-50bx^{2}$$

Evaluate

$$5b\left(5-2b-b^{2}\right),\ x^{2}\left(2x^{2}+x+2\right)$$