Solve x^2-16/x^2-9*2x(x-9)+3(x-9)/2x^2-11x+12div4x+6/x+9

$$\frac { x ^ { 2 } - 16 } { x ^ { 2 } - 9 } \times \frac { 2 x ( x - 9 ) + 3 ( x - 9 ) } { 2 x ^ { 2 } - 11 x + 12 } \div \frac { 4 x + 6 } { x + 9 }$$

Evaluate

$$\frac{\left(x+4\right)\left(x^{2}-81\right)}{2\left(2x-3\right)\left(x^{2}-9\right)}$$

Short Solution Steps

Multiply \frac{x^{2}-16}{x^{2}-9} times \frac{2x\left(x-9\right)+3\left(x-9\right)}{2x^{2}-11x+12} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{\left(x^{2}-16\right)\left(2x\left(x-9\right)+3\left(x-9\right)\right)}{\left(x^{2}-9\right)\left(2x^{2}-11x+12\right)}}{\frac{4x+6}{x+9}}$$

Divide \frac{\left(x^{2}-16\right)\left(2x\left(x-9\right)+3\left(x-9\right)\right)}{\left(x^{2}-9\right)\left(2x^{2}-11x+12\right)} by \frac{4x+6}{x+9} by multiplying \frac{\left(x^{2}-16\right)\left(2x\left(x-9\right)+3\left(x-9\right)\right)}{\left(x^{2}-9\right)\left(2x^{2}-11x+12\right)} by the reciprocal of \frac{4x+6}{x+9}.

$$\frac{\left(x^{2}-16\right)\left(2x\left(x-9\right)+3\left(x-9\right)\right)\left(x+9\right)}{\left(x^{2}-9\right)\left(2x^{2}-11x+12\right)\left(4x+6\right)}$$

Factor the expressions that are not already factored.

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

Cancel out \left(x-4\right)\left(2x+3\right) in both numerator and denominator.

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

Expand the expression.

$$\frac{x^{3}+4x^{2}-81x-324}{4x^{3}-6x^{2}-36x+54}$$

Expand

$$\frac{x^{3}+4x^{2}-81x-324}{2\left(2x-3\right)\left(x^{2}-9\right)}$$

Short Solution Steps

Multiply \frac{x^{2}-16}{x^{2}-9} times \frac{2x\left(x-9\right)+3\left(x-9\right)}{2x^{2}-11x+12} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{\left(x^{2}-16\right)\left(2x\left(x-9\right)+3\left(x-9\right)\right)}{\left(x^{2}-9\right)\left(2x^{2}-11x+12\right)}}{\frac{4x+6}{x+9}}$$

Divide \frac{\left(x^{2}-16\right)\left(2x\left(x-9\right)+3\left(x-9\right)\right)}{\left(x^{2}-9\right)\left(2x^{2}-11x+12\right)} by \frac{4x+6}{x+9} by multiplying \frac{\left(x^{2}-16\right)\left(2x\left(x-9\right)+3\left(x-9\right)\right)}{\left(x^{2}-9\right)\left(2x^{2}-11x+12\right)} by the reciprocal of \frac{4x+6}{x+9}.

$$\frac{\left(x^{2}-16\right)\left(2x\left(x-9\right)+3\left(x-9\right)\right)\left(x+9\right)}{\left(x^{2}-9\right)\left(2x^{2}-11x+12\right)\left(4x+6\right)}$$

Factor the expressions that are not already factored.

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

Cancel out \left(x-4\right)\left(2x+3\right) in both numerator and denominator.

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

Expand the expression.

$$\frac{x^{3}+4x^{2}-81x-324}{4x^{3}-6x^{2}-36x+54}$$