Solve [10/x^2+2x-15+x+1/9-3x+x/2x+10]*x-9/x

$$[ \frac { 10 } { x ^ { 2 } + 2 x - 15 } + \frac { x + 1 } { 9 - 3 x } + \frac { x } { 2 x + 10 } ] \cdot \frac { x - 9 } { x }$$

Evaluate

$$\frac{\left(x-9\right)\left(-x^{2}+21x-50\right)}{6x\left(3-x\right)\left(x+5\right)}$$

Short Solution Steps

Factor x^{2}+2x-15. Factor 9-3x.

$$\left(\frac{10}{\left(x-3\right)\left(x+5\right)}+\frac{x+1}{3\left(-x+3\right)}+\frac{x}{2x+10}\right)\times \frac{x-9}{x}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-3\right)\left(x+5\right) and 3\left(-x+3\right) is 3\left(x-3\right)\left(x+5\right). Multiply \frac{10}{\left(x-3\right)\left(x+5\right)} times \frac{3}{3}. Multiply \frac{x+1}{3\left(-x+3\right)} times \frac{-\left(x+5\right)}{-\left(x+5\right)}.

$$\left(\frac{10\times 3}{3\left(x-3\right)\left(x+5\right)}+\frac{\left(x+1\right)\left(-1\right)\left(x+5\right)}{3\left(x-3\right)\left(x+5\right)}+\frac{x}{2x+10}\right)\times \frac{x-9}{x}$$

Since \frac{10\times 3}{3\left(x-3\right)\left(x+5\right)} and \frac{\left(x+1\right)\left(-1\right)\left(x+5\right)}{3\left(x-3\right)\left(x+5\right)} have the same denominator, add them by adding their numerators.

$$\left(\frac{10\times 3+\left(x+1\right)\left(-1\right)\left(x+5\right)}{3\left(x-3\right)\left(x+5\right)}+\frac{x}{2x+10}\right)\times \frac{x-9}{x}$$

Do the multiplications in 10\times 3+\left(x+1\right)\left(-1\right)\left(x+5\right).

$$\left(\frac{30-x^{2}-5x-x-5}{3\left(x-3\right)\left(x+5\right)}+\frac{x}{2x+10}\right)\times \frac{x-9}{x}$$

Combine like terms in 30-x^{2}-5x-x-5.

$$\left(\frac{25-x^{2}-6x}{3\left(x-3\right)\left(x+5\right)}+\frac{x}{2x+10}\right)\times \frac{x-9}{x}$$

Factor 2x+10.

$$\left(\frac{25-x^{2}-6x}{3\left(x-3\right)\left(x+5\right)}+\frac{x}{2\left(x+5\right)}\right)\times \frac{x-9}{x}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3\left(x-3\right)\left(x+5\right) and 2\left(x+5\right) is 6\left(x-3\right)\left(x+5\right). Multiply \frac{25-x^{2}-6x}{3\left(x-3\right)\left(x+5\right)} times \frac{2}{2}. Multiply \frac{x}{2\left(x+5\right)} times \frac{3\left(x-3\right)}{3\left(x-3\right)}.

$$\left(\frac{2\left(25-x^{2}-6x\right)}{6\left(x-3\right)\left(x+5\right)}+\frac{x\times 3\left(x-3\right)}{6\left(x-3\right)\left(x+5\right)}\right)\times \frac{x-9}{x}$$

Since \frac{2\left(25-x^{2}-6x\right)}{6\left(x-3\right)\left(x+5\right)} and \frac{x\times 3\left(x-3\right)}{6\left(x-3\right)\left(x+5\right)} have the same denominator, add them by adding their numerators.

$$\frac{2\left(25-x^{2}-6x\right)+x\times 3\left(x-3\right)}{6\left(x-3\right)\left(x+5\right)}\times \frac{x-9}{x}$$

Do the multiplications in 2\left(25-x^{2}-6x\right)+x\times 3\left(x-3\right).

$$\frac{50-2x^{2}-12x+3x^{2}-9x}{6\left(x-3\right)\left(x+5\right)}\times \frac{x-9}{x}$$

Combine like terms in 50-2x^{2}-12x+3x^{2}-9x.

$$\frac{50+x^{2}-21x}{6\left(x-3\right)\left(x+5\right)}\times \frac{x-9}{x}$$

Multiply \frac{50+x^{2}-21x}{6\left(x-3\right)\left(x+5\right)} times \frac{x-9}{x} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(50+x^{2}-21x\right)\left(x-9\right)}{6\left(x-3\right)\left(x+5\right)x}$$

Use the distributive property to multiply 50+x^{2}-21x by x-9 and combine like terms.

$$\frac{239x-450+x^{3}-30x^{2}}{6\left(x-3\right)\left(x+5\right)x}$$

Use the distributive property to multiply 6 by x-3.

$$\frac{239x-450+x^{3}-30x^{2}}{\left(6x-18\right)\left(x+5\right)x}$$

Use the distributive property to multiply 6x-18 by x+5 and combine like terms.

$$\frac{239x-450+x^{3}-30x^{2}}{\left(6x^{2}+12x-90\right)x}$$

Use the distributive property to multiply 6x^{2}+12x-90 by x.

$$\frac{239x-450+x^{3}-30x^{2}}{6x^{3}+12x^{2}-90x}$$

Expand

$$\frac{x^{3}-30x^{2}+239x-450}{6x\left(x-3\right)\left(x+5\right)}$$

Short Solution Steps

Factor x^{2}+2x-15. Factor 9-3x.

$$\left(\frac{10}{\left(x-3\right)\left(x+5\right)}+\frac{x+1}{3\left(-x+3\right)}+\frac{x}{2x+10}\right)\times \frac{x-9}{x}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-3\right)\left(x+5\right) and 3\left(-x+3\right) is 3\left(x-3\right)\left(x+5\right). Multiply \frac{10}{\left(x-3\right)\left(x+5\right)} times \frac{3}{3}. Multiply \frac{x+1}{3\left(-x+3\right)} times \frac{-\left(x+5\right)}{-\left(x+5\right)}.

$$\left(\frac{10\times 3}{3\left(x-3\right)\left(x+5\right)}+\frac{\left(x+1\right)\left(-1\right)\left(x+5\right)}{3\left(x-3\right)\left(x+5\right)}+\frac{x}{2x+10}\right)\times \frac{x-9}{x}$$

Since \frac{10\times 3}{3\left(x-3\right)\left(x+5\right)} and \frac{\left(x+1\right)\left(-1\right)\left(x+5\right)}{3\left(x-3\right)\left(x+5\right)} have the same denominator, add them by adding their numerators.

$$\left(\frac{10\times 3+\left(x+1\right)\left(-1\right)\left(x+5\right)}{3\left(x-3\right)\left(x+5\right)}+\frac{x}{2x+10}\right)\times \frac{x-9}{x}$$

Do the multiplications in 10\times 3+\left(x+1\right)\left(-1\right)\left(x+5\right).

$$\left(\frac{30-x^{2}-5x-x-5}{3\left(x-3\right)\left(x+5\right)}+\frac{x}{2x+10}\right)\times \frac{x-9}{x}$$

Combine like terms in 30-x^{2}-5x-x-5.

$$\left(\frac{25-x^{2}-6x}{3\left(x-3\right)\left(x+5\right)}+\frac{x}{2x+10}\right)\times \frac{x-9}{x}$$

Factor 2x+10.

$$\left(\frac{25-x^{2}-6x}{3\left(x-3\right)\left(x+5\right)}+\frac{x}{2\left(x+5\right)}\right)\times \frac{x-9}{x}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3\left(x-3\right)\left(x+5\right) and 2\left(x+5\right) is 6\left(x-3\right)\left(x+5\right). Multiply \frac{25-x^{2}-6x}{3\left(x-3\right)\left(x+5\right)} times \frac{2}{2}. Multiply \frac{x}{2\left(x+5\right)} times \frac{3\left(x-3\right)}{3\left(x-3\right)}.

$$\left(\frac{2\left(25-x^{2}-6x\right)}{6\left(x-3\right)\left(x+5\right)}+\frac{x\times 3\left(x-3\right)}{6\left(x-3\right)\left(x+5\right)}\right)\times \frac{x-9}{x}$$

Since \frac{2\left(25-x^{2}-6x\right)}{6\left(x-3\right)\left(x+5\right)} and \frac{x\times 3\left(x-3\right)}{6\left(x-3\right)\left(x+5\right)} have the same denominator, add them by adding their numerators.

$$\frac{2\left(25-x^{2}-6x\right)+x\times 3\left(x-3\right)}{6\left(x-3\right)\left(x+5\right)}\times \frac{x-9}{x}$$

Do the multiplications in 2\left(25-x^{2}-6x\right)+x\times 3\left(x-3\right).

$$\frac{50-2x^{2}-12x+3x^{2}-9x}{6\left(x-3\right)\left(x+5\right)}\times \frac{x-9}{x}$$

Combine like terms in 50-2x^{2}-12x+3x^{2}-9x.

$$\frac{50+x^{2}-21x}{6\left(x-3\right)\left(x+5\right)}\times \frac{x-9}{x}$$

Multiply \frac{50+x^{2}-21x}{6\left(x-3\right)\left(x+5\right)} times \frac{x-9}{x} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(50+x^{2}-21x\right)\left(x-9\right)}{6\left(x-3\right)\left(x+5\right)x}$$

Use the distributive property to multiply 50+x^{2}-21x by x-9 and combine like terms.

$$\frac{239x-450+x^{3}-30x^{2}}{6\left(x-3\right)\left(x+5\right)x}$$

Use the distributive property to multiply 6 by x-3.

$$\frac{239x-450+x^{3}-30x^{2}}{\left(6x-18\right)\left(x+5\right)x}$$

Use the distributive property to multiply 6x-18 by x+5 and combine like terms.

$$\frac{239x-450+x^{3}-30x^{2}}{\left(6x^{2}+12x-90\right)x}$$

Use the distributive property to multiply 6x^{2}+12x-90 by x.

$$\frac{239x-450+x^{3}-30x^{2}}{6x^{3}+12x^{2}-90x}$$