Solve frac{x^{2}+5x+6}{x-5}*25-x^2/x^2+7x+10div6x^3-54x/3x^2-27

$$\frac { x ^ { 2 } + 5 x + 6 } { x - 5 } \cdot \frac { 25 - x ^ { 2 } } { x ^ { 2 } + 7 x + 10 } \div \frac { 6 x ^ { 3 } - 54 x } { 3 x ^ { 2 } - 27 }$$

Evaluate

$$-\frac{1}{2}-\frac{3}{2x}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{25-x^{2}}{x^{2}+7x+10}.

$$\frac{\frac{x^{2}+5x+6}{x-5}\times \frac{\left(x-5\right)\left(-x-5\right)}{\left(x+2\right)\left(x+5\right)}}{\frac{6x^{3}-54x}{3x^{2}-27}}$$

Extract the negative sign in -5-x.

$$\frac{\frac{x^{2}+5x+6}{x-5}\times \frac{-\left(x-5\right)\left(x+5\right)}{\left(x+2\right)\left(x+5\right)}}{\frac{6x^{3}-54x}{3x^{2}-27}}$$

Cancel out x+5 in both numerator and denominator.

$$\frac{\frac{x^{2}+5x+6}{x-5}\times \frac{-\left(x-5\right)}{x+2}}{\frac{6x^{3}-54x}{3x^{2}-27}}$$

Multiply \frac{x^{2}+5x+6}{x-5} times \frac{-\left(x-5\right)}{x+2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{\left(x^{2}+5x+6\right)\left(-1\right)\left(x-5\right)}{\left(x-5\right)\left(x+2\right)}}{\frac{6x^{3}-54x}{3x^{2}-27}}$$

Cancel out x-5 in both numerator and denominator.

$$\frac{\frac{-\left(x^{2}+5x+6\right)}{x+2}}{\frac{6x^{3}-54x}{3x^{2}-27}}$$

Factor the expressions that are not already factored in \frac{6x^{3}-54x}{3x^{2}-27}.

$$\frac{\frac{-\left(x^{2}+5x+6\right)}{x+2}}{\frac{6x\left(x-3\right)\left(x+3\right)}{3\left(x-3\right)\left(x+3\right)}}$$

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

$$\frac{\frac{-\left(x^{2}+5x+6\right)}{x+2}}{2x}$$

Express \frac{\frac{-\left(x^{2}+5x+6\right)}{x+2}}{2x} as a single fraction.

$$\frac{-\left(x^{2}+5x+6\right)}{\left(x+2\right)\times 2x}$$

Factor the expressions that are not already factored.

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

Cancel out x+2 in both numerator and denominator.

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

Expand the expression.

$$\frac{-x-3}{2x}$$

Expand

$$-\frac{1}{2}-\frac{3}{2x}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{25-x^{2}}{x^{2}+7x+10}.

$$\frac{\frac{x^{2}+5x+6}{x-5}\times \frac{\left(x-5\right)\left(-x-5\right)}{\left(x+2\right)\left(x+5\right)}}{\frac{6x^{3}-54x}{3x^{2}-27}}$$

Extract the negative sign in -5-x.

$$\frac{\frac{x^{2}+5x+6}{x-5}\times \frac{-\left(x-5\right)\left(x+5\right)}{\left(x+2\right)\left(x+5\right)}}{\frac{6x^{3}-54x}{3x^{2}-27}}$$

Cancel out x+5 in both numerator and denominator.

$$\frac{\frac{x^{2}+5x+6}{x-5}\times \frac{-\left(x-5\right)}{x+2}}{\frac{6x^{3}-54x}{3x^{2}-27}}$$

Multiply \frac{x^{2}+5x+6}{x-5} times \frac{-\left(x-5\right)}{x+2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{\left(x^{2}+5x+6\right)\left(-1\right)\left(x-5\right)}{\left(x-5\right)\left(x+2\right)}}{\frac{6x^{3}-54x}{3x^{2}-27}}$$

Cancel out x-5 in both numerator and denominator.

$$\frac{\frac{-\left(x^{2}+5x+6\right)}{x+2}}{\frac{6x^{3}-54x}{3x^{2}-27}}$$

Factor the expressions that are not already factored in \frac{6x^{3}-54x}{3x^{2}-27}.

$$\frac{\frac{-\left(x^{2}+5x+6\right)}{x+2}}{\frac{6x\left(x-3\right)\left(x+3\right)}{3\left(x-3\right)\left(x+3\right)}}$$

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

$$\frac{\frac{-\left(x^{2}+5x+6\right)}{x+2}}{2x}$$

Express \frac{\frac{-\left(x^{2}+5x+6\right)}{x+2}}{2x} as a single fraction.

$$\frac{-\left(x^{2}+5x+6\right)}{\left(x+2\right)\times 2x}$$

Factor the expressions that are not already factored.

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

Cancel out x+2 in both numerator and denominator.

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

Expand the expression.

$$\frac{-x-3}{2x}$$