Solve frac{x^{2}+9}{x^{2}-9}*2x+6/x^2+6x+9:2x^2+18/x^2-5x+6+2/x^2+3x

$$\frac { x ^ { 2 } + 9 } { x ^ { 2 } - 9 } \cdot \frac { 2 x + 6 } { x ^ { 2 } + 6 x + 9 } : \frac { 2 x ^ { 2 } + 18 } { x ^ { 2 } - 5 x + 6 } + \frac { 2 } { x ^ { 2 } + 3 x }$$

Evaluate

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{2x+6}{x^{2}+6x+9}.

$$\frac{\frac{x^{2}+9}{x^{2}-9}\times \frac{2\left(x+3\right)}{\left(x+3\right)^{2}}}{\frac{2x^{2}+18}{x^{2}-5x+6}}+\frac{2}{x^{2}+3x}$$

Cancel out x+3 in both numerator and denominator.

$$\frac{\frac{x^{2}+9}{x^{2}-9}\times \frac{2}{x+3}}{\frac{2x^{2}+18}{x^{2}-5x+6}}+\frac{2}{x^{2}+3x}$$

Multiply \frac{x^{2}+9}{x^{2}-9} times \frac{2}{x+3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{\left(x^{2}+9\right)\times 2}{\left(x^{2}-9\right)\left(x+3\right)}}{\frac{2x^{2}+18}{x^{2}-5x+6}}+\frac{2}{x^{2}+3x}$$

Divide \frac{\left(x^{2}+9\right)\times 2}{\left(x^{2}-9\right)\left(x+3\right)} by \frac{2x^{2}+18}{x^{2}-5x+6} by multiplying \frac{\left(x^{2}+9\right)\times 2}{\left(x^{2}-9\right)\left(x+3\right)} by the reciprocal of \frac{2x^{2}+18}{x^{2}-5x+6}.

$$\frac{\left(x^{2}+9\right)\times 2\left(x^{2}-5x+6\right)}{\left(x^{2}-9\right)\left(x+3\right)\left(2x^{2}+18\right)}+\frac{2}{x^{2}+3x}$$

Factor the expressions that are not already factored in \frac{\left(x^{2}+9\right)\times 2\left(x^{2}-5x+6\right)}{\left(x^{2}-9\right)\left(x+3\right)\left(2x^{2}+18\right)}.

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

Cancel out 2\left(x-3\right)\left(x^{2}+9\right) in both numerator and denominator.

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

Factor x^{2}+3x.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+3\right)^{2} and x\left(x+3\right) is x\left(x+3\right)^{2}. Multiply \frac{x-2}{\left(x+3\right)^{2}} times \frac{x}{x}. Multiply \frac{2}{x\left(x+3\right)} times \frac{x+3}{x+3}.

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

Since \frac{\left(x-2\right)x}{x\left(x+3\right)^{2}} and \frac{2\left(x+3\right)}{x\left(x+3\right)^{2}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(x-2\right)x+2\left(x+3\right).

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

Combine like terms in x^{2}-2x+2x+6.

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

Expand x\left(x+3\right)^{2}.

$$\frac{x^{2}+6}{x^{3}+6x^{2}+9x}$$

Expand

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{2x+6}{x^{2}+6x+9}.

$$\frac{\frac{x^{2}+9}{x^{2}-9}\times \frac{2\left(x+3\right)}{\left(x+3\right)^{2}}}{\frac{2x^{2}+18}{x^{2}-5x+6}}+\frac{2}{x^{2}+3x}$$

Cancel out x+3 in both numerator and denominator.

$$\frac{\frac{x^{2}+9}{x^{2}-9}\times \frac{2}{x+3}}{\frac{2x^{2}+18}{x^{2}-5x+6}}+\frac{2}{x^{2}+3x}$$

Multiply \frac{x^{2}+9}{x^{2}-9} times \frac{2}{x+3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{\left(x^{2}+9\right)\times 2}{\left(x^{2}-9\right)\left(x+3\right)}}{\frac{2x^{2}+18}{x^{2}-5x+6}}+\frac{2}{x^{2}+3x}$$

Divide \frac{\left(x^{2}+9\right)\times 2}{\left(x^{2}-9\right)\left(x+3\right)} by \frac{2x^{2}+18}{x^{2}-5x+6} by multiplying \frac{\left(x^{2}+9\right)\times 2}{\left(x^{2}-9\right)\left(x+3\right)} by the reciprocal of \frac{2x^{2}+18}{x^{2}-5x+6}.

$$\frac{\left(x^{2}+9\right)\times 2\left(x^{2}-5x+6\right)}{\left(x^{2}-9\right)\left(x+3\right)\left(2x^{2}+18\right)}+\frac{2}{x^{2}+3x}$$

Factor the expressions that are not already factored in \frac{\left(x^{2}+9\right)\times 2\left(x^{2}-5x+6\right)}{\left(x^{2}-9\right)\left(x+3\right)\left(2x^{2}+18\right)}.

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

Cancel out 2\left(x-3\right)\left(x^{2}+9\right) in both numerator and denominator.

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

Factor x^{2}+3x.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+3\right)^{2} and x\left(x+3\right) is x\left(x+3\right)^{2}. Multiply \frac{x-2}{\left(x+3\right)^{2}} times \frac{x}{x}. Multiply \frac{2}{x\left(x+3\right)} times \frac{x+3}{x+3}.

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

Since \frac{\left(x-2\right)x}{x\left(x+3\right)^{2}} and \frac{2\left(x+3\right)}{x\left(x+3\right)^{2}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(x-2\right)x+2\left(x+3\right).

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

Combine like terms in x^{2}-2x+2x+6.

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

Expand x\left(x+3\right)^{2}.

$$\frac{x^{2}+6}{x^{3}+6x^{2}+9x}$$