Solve R(x)=4x^2-9x-9/x^3-8/frac{x^2-6x+9{x^2-4}}

$$R ( x ) = \frac { \frac { 4 x ^ { 2 } - 9 x - 9 } { x ^ { 3 } - 8 } } { \frac { x ^ { 2 } - 6 x + 9 } { x ^ { 2 } - 4 } }$$

Evaluate

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

Short Solution Steps

Divide \frac{4x^{2}-9x-9}{x^{3}-8} by \frac{x^{2}-6x+9}{x^{2}-4} by multiplying \frac{4x^{2}-9x-9}{x^{3}-8} by the reciprocal of \frac{x^{2}-6x+9}{x^{2}-4}.

$$\frac{\left(4x^{2}-9x-9\right)\left(x^{2}-4\right)}{\left(x^{3}-8\right)\left(x^{2}-6x+9\right)}$$

Factor the expressions that are not already factored.

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

Cancel out \left(x-3\right)\left(x-2\right) in both numerator and denominator.

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

Expand the expression.

$$\frac{4x^{2}+11x+6}{x^{3}-x^{2}-2x-12}$$

Expand

$$\frac{4x^{2}+11x+6}{\left(x-3\right)\left(x^{2}+2x+4\right)}$$

Short Solution Steps

Divide \frac{4x^{2}-9x-9}{x^{3}-8} by \frac{x^{2}-6x+9}{x^{2}-4} by multiplying \frac{4x^{2}-9x-9}{x^{3}-8} by the reciprocal of \frac{x^{2}-6x+9}{x^{2}-4}.

$$\frac{\left(4x^{2}-9x-9\right)\left(x^{2}-4\right)}{\left(x^{3}-8\right)\left(x^{2}-6x+9\right)}$$

Factor the expressions that are not already factored.

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

Cancel out \left(x-3\right)\left(x-2\right) in both numerator and denominator.

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

Expand the expression.

$$\frac{4x^{2}+11x+6}{x^{3}-x^{2}-2x-12}$$