Solve x^4+2x^3-5x^2-18x-36div(x^2-9)

$${ x }^{ 4 } +2 { x }^{ 3 } -5 { x }^{ 2 } -18x-36 \div ( { x }^{ 2 } -9)$$

Evaluate

$$\frac{x^{6}+2x^{5}-14x^{4}-36x^{3}+45x^{2}+162x-36}{x^{2}-9}$$

Short Solution Steps

Factor x^{2}-9.

$$x^{4}+2x^{3}-5x^{2}-18x-\frac{36}{\left(x-3\right)\left(x+3\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x^{4}+2x^{3}-5x^{2}-18x times \frac{\left(x-3\right)\left(x+3\right)}{\left(x-3\right)\left(x+3\right)}.

$$\frac{\left(x^{4}+2x^{3}-5x^{2}-18x\right)\left(x-3\right)\left(x+3\right)}{\left(x-3\right)\left(x+3\right)}-\frac{36}{\left(x-3\right)\left(x+3\right)}$$

Since \frac{\left(x^{4}+2x^{3}-5x^{2}-18x\right)\left(x-3\right)\left(x+3\right)}{\left(x-3\right)\left(x+3\right)} and \frac{36}{\left(x-3\right)\left(x+3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x^{4}+2x^{3}-5x^{2}-18x\right)\left(x-3\right)\left(x+3\right)-36}{\left(x-3\right)\left(x+3\right)}$$

Do the multiplications in \left(x^{4}+2x^{3}-5x^{2}-18x\right)\left(x-3\right)\left(x+3\right)-36.

$$\frac{x^{6}-9x^{4}+2x^{5}-18x^{3}-5x^{4}+45x^{2}-18x^{3}+162x-36}{\left(x-3\right)\left(x+3\right)}$$

Combine like terms in x^{6}-9x^{4}+2x^{5}-18x^{3}-5x^{4}+45x^{2}-18x^{3}+162x-36.

$$\frac{162x+x^{6}-14x^{4}+2x^{5}-36x^{3}+45x^{2}-36}{\left(x-3\right)\left(x+3\right)}$$

Expand \left(x-3\right)\left(x+3\right).

$$\frac{162x+x^{6}-14x^{4}+2x^{5}-36x^{3}+45x^{2}-36}{x^{2}-9}$$

Differentiate w.r.t. x

$$\frac{2\left(2x^{7}+3x^{6}-41x^{5}-63x^{4}+252x^{3}+405x^{2}-369x-729\right)}{\left(x^{2}-9\right)^{2}}$$