Solve -2x^2+18div(x^4+18x^2+81)

$$-2 { x }^{ 2 } +18 \div ( { x }^{ 4 } +18 { x }^{ 2 } +81)$$

Evaluate

$$\frac{2\left(-\left(x^{3}+9x\right)^{2}+9\right)}{\left(x^{2}+9\right)^{2}}$$

Short Solution Steps

Factor x^{4}+18x^{2}+81.

$$-2x^{2}+\frac{18}{\left(x^{2}+9\right)^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -2x^{2} times \frac{\left(x^{2}+9\right)^{2}}{\left(x^{2}+9\right)^{2}}.

$$\frac{-2x^{2}\left(x^{2}+9\right)^{2}}{\left(x^{2}+9\right)^{2}}+\frac{18}{\left(x^{2}+9\right)^{2}}$$

Since \frac{-2x^{2}\left(x^{2}+9\right)^{2}}{\left(x^{2}+9\right)^{2}} and \frac{18}{\left(x^{2}+9\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{-2x^{2}\left(x^{2}+9\right)^{2}+18}{\left(x^{2}+9\right)^{2}}$$

Do the multiplications in -2x^{2}\left(x^{2}+9\right)^{2}+18.

$$\frac{-2x^{6}-36x^{4}-162x^{2}+18}{\left(x^{2}+9\right)^{2}}$$

Expand \left(x^{2}+9\right)^{2}.

$$\frac{-2x^{6}-36x^{4}-162x^{2}+18}{x^{4}+18x^{2}+81}$$

Factor

$$-\frac{2\left(x^{3}+9x-3\right)\left(x^{3}+9x+3\right)}{\left(x^{2}+9\right)^{2}}$$