Solve -4/(x+3)^2+-10/3/x+3+3/x+2+frac{frac{1}{3}}{x}

$$\frac { - 4 } { ( x + 3 ) ^ { 2 } } + \frac { - \frac { 10 } { 3 } } { x + 3 } + \frac { 3 } { x + 2 } + \frac { \frac { 1 } { 3 } } { x }$$

Evaluate

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

Short Solution Steps

Express \frac{-\frac{10}{3}}{x+3} as a single fraction.

$$\frac{-4}{\left(x+3\right)^{2}}+\frac{-10}{3\left(x+3\right)}+\frac{3}{x+2}+\frac{\frac{1}{3}}{x}$$

Express \frac{\frac{1}{3}}{x} as a single fraction.

$$\frac{-4}{\left(x+3\right)^{2}}+\frac{-10}{3\left(x+3\right)}+\frac{3}{x+2}+\frac{1}{3x}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+3\right)^{2} and 3\left(x+3\right) is 3\left(x+3\right)^{2}. Multiply \frac{-4}{\left(x+3\right)^{2}} times \frac{3}{3}. Multiply \frac{-10}{3\left(x+3\right)} times \frac{x+3}{x+3}.

$$\frac{-4\times 3}{3\left(x+3\right)^{2}}+\frac{-10\left(x+3\right)}{3\left(x+3\right)^{2}}+\frac{3}{x+2}+\frac{1}{3x}$$

Since \frac{-4\times 3}{3\left(x+3\right)^{2}} and \frac{-10\left(x+3\right)}{3\left(x+3\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{-4\times 3-10\left(x+3\right)}{3\left(x+3\right)^{2}}+\frac{3}{x+2}+\frac{1}{3x}$$

Do the multiplications in -4\times 3-10\left(x+3\right).

$$\frac{-12-10x-30}{3\left(x+3\right)^{2}}+\frac{3}{x+2}+\frac{1}{3x}$$

Combine like terms in -12-10x-30.

$$\frac{-42-10x}{3\left(x+3\right)^{2}}+\frac{3}{x+2}+\frac{1}{3x}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3\left(x+3\right)^{2} and x+2 is 3\left(x+2\right)\left(x+3\right)^{2}. Multiply \frac{-42-10x}{3\left(x+3\right)^{2}} times \frac{x+2}{x+2}. Multiply \frac{3}{x+2} times \frac{3\left(x+3\right)^{2}}{3\left(x+3\right)^{2}}.

$$\frac{\left(-42-10x\right)\left(x+2\right)}{3\left(x+2\right)\left(x+3\right)^{2}}+\frac{3\times 3\left(x+3\right)^{2}}{3\left(x+2\right)\left(x+3\right)^{2}}+\frac{1}{3x}$$

Since \frac{\left(-42-10x\right)\left(x+2\right)}{3\left(x+2\right)\left(x+3\right)^{2}} and \frac{3\times 3\left(x+3\right)^{2}}{3\left(x+2\right)\left(x+3\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(-42-10x\right)\left(x+2\right)+3\times 3\left(x+3\right)^{2}}{3\left(x+2\right)\left(x+3\right)^{2}}+\frac{1}{3x}$$

Do the multiplications in \left(-42-10x\right)\left(x+2\right)+3\times 3\left(x+3\right)^{2}.

$$\frac{-42x-84-10x^{2}-20x+9x^{2}+54x+81}{3\left(x+2\right)\left(x+3\right)^{2}}+\frac{1}{3x}$$

Combine like terms in -42x-84-10x^{2}-20x+9x^{2}+54x+81.

$$\frac{-8x-3-x^{2}}{3\left(x+2\right)\left(x+3\right)^{2}}+\frac{1}{3x}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3\left(x+2\right)\left(x+3\right)^{2} and 3x is 3x\left(x+2\right)\left(x+3\right)^{2}. Multiply \frac{-8x-3-x^{2}}{3\left(x+2\right)\left(x+3\right)^{2}} times \frac{x}{x}. Multiply \frac{1}{3x} times \frac{\left(x+2\right)\left(x+3\right)^{2}}{\left(x+2\right)\left(x+3\right)^{2}}.

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

Since \frac{\left(-8x-3-x^{2}\right)x}{3x\left(x+2\right)\left(x+3\right)^{2}} and \frac{\left(x+2\right)\left(x+3\right)^{2}}{3x\left(x+2\right)\left(x+3\right)^{2}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(-8x-3-x^{2}\right)x+\left(x+2\right)\left(x+3\right)^{2}.

$$\frac{-8x^{2}-3x-x^{3}+x^{3}+6x^{2}+9x+2x^{2}+12x+18}{3x\left(x+2\right)\left(x+3\right)^{2}}$$

Combine like terms in -8x^{2}-3x-x^{3}+x^{3}+6x^{2}+9x+2x^{2}+12x+18.

$$\frac{18x+18}{3x\left(x+2\right)\left(x+3\right)^{2}}$$

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

$$\frac{18\left(x+1\right)}{3x\left(x+2\right)\left(x+3\right)^{2}}$$

Cancel out 3 in both numerator and denominator.

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

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

$$\frac{6\left(x+1\right)}{x^{4}+8x^{3}+21x^{2}+18x}$$

Use the distributive property to multiply 6 by x+1.

$$\frac{6x+6}{x^{4}+8x^{3}+21x^{2}+18x}$$

Expand

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

Short Solution Steps

Express \frac{-\frac{10}{3}}{x+3} as a single fraction.

$$\frac{-4}{\left(x+3\right)^{2}}+\frac{-10}{3\left(x+3\right)}+\frac{3}{x+2}+\frac{\frac{1}{3}}{x}$$

Express \frac{\frac{1}{3}}{x} as a single fraction.

$$\frac{-4}{\left(x+3\right)^{2}}+\frac{-10}{3\left(x+3\right)}+\frac{3}{x+2}+\frac{1}{3x}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+3\right)^{2} and 3\left(x+3\right) is 3\left(x+3\right)^{2}. Multiply \frac{-4}{\left(x+3\right)^{2}} times \frac{3}{3}. Multiply \frac{-10}{3\left(x+3\right)} times \frac{x+3}{x+3}.

$$\frac{-4\times 3}{3\left(x+3\right)^{2}}+\frac{-10\left(x+3\right)}{3\left(x+3\right)^{2}}+\frac{3}{x+2}+\frac{1}{3x}$$

Since \frac{-4\times 3}{3\left(x+3\right)^{2}} and \frac{-10\left(x+3\right)}{3\left(x+3\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{-4\times 3-10\left(x+3\right)}{3\left(x+3\right)^{2}}+\frac{3}{x+2}+\frac{1}{3x}$$

Do the multiplications in -4\times 3-10\left(x+3\right).

$$\frac{-12-10x-30}{3\left(x+3\right)^{2}}+\frac{3}{x+2}+\frac{1}{3x}$$

Combine like terms in -12-10x-30.

$$\frac{-42-10x}{3\left(x+3\right)^{2}}+\frac{3}{x+2}+\frac{1}{3x}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3\left(x+3\right)^{2} and x+2 is 3\left(x+2\right)\left(x+3\right)^{2}. Multiply \frac{-42-10x}{3\left(x+3\right)^{2}} times \frac{x+2}{x+2}. Multiply \frac{3}{x+2} times \frac{3\left(x+3\right)^{2}}{3\left(x+3\right)^{2}}.

$$\frac{\left(-42-10x\right)\left(x+2\right)}{3\left(x+2\right)\left(x+3\right)^{2}}+\frac{3\times 3\left(x+3\right)^{2}}{3\left(x+2\right)\left(x+3\right)^{2}}+\frac{1}{3x}$$

Since \frac{\left(-42-10x\right)\left(x+2\right)}{3\left(x+2\right)\left(x+3\right)^{2}} and \frac{3\times 3\left(x+3\right)^{2}}{3\left(x+2\right)\left(x+3\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(-42-10x\right)\left(x+2\right)+3\times 3\left(x+3\right)^{2}}{3\left(x+2\right)\left(x+3\right)^{2}}+\frac{1}{3x}$$

Do the multiplications in \left(-42-10x\right)\left(x+2\right)+3\times 3\left(x+3\right)^{2}.

$$\frac{-42x-84-10x^{2}-20x+9x^{2}+54x+81}{3\left(x+2\right)\left(x+3\right)^{2}}+\frac{1}{3x}$$

Combine like terms in -42x-84-10x^{2}-20x+9x^{2}+54x+81.

$$\frac{-8x-3-x^{2}}{3\left(x+2\right)\left(x+3\right)^{2}}+\frac{1}{3x}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3\left(x+2\right)\left(x+3\right)^{2} and 3x is 3x\left(x+2\right)\left(x+3\right)^{2}. Multiply \frac{-8x-3-x^{2}}{3\left(x+2\right)\left(x+3\right)^{2}} times \frac{x}{x}. Multiply \frac{1}{3x} times \frac{\left(x+2\right)\left(x+3\right)^{2}}{\left(x+2\right)\left(x+3\right)^{2}}.

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

Since \frac{\left(-8x-3-x^{2}\right)x}{3x\left(x+2\right)\left(x+3\right)^{2}} and \frac{\left(x+2\right)\left(x+3\right)^{2}}{3x\left(x+2\right)\left(x+3\right)^{2}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(-8x-3-x^{2}\right)x+\left(x+2\right)\left(x+3\right)^{2}.

$$\frac{-8x^{2}-3x-x^{3}+x^{3}+6x^{2}+9x+2x^{2}+12x+18}{3x\left(x+2\right)\left(x+3\right)^{2}}$$

Combine like terms in -8x^{2}-3x-x^{3}+x^{3}+6x^{2}+9x+2x^{2}+12x+18.

$$\frac{18x+18}{3x\left(x+2\right)\left(x+3\right)^{2}}$$

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

$$\frac{18\left(x+1\right)}{3x\left(x+2\right)\left(x+3\right)^{2}}$$

Cancel out 3 in both numerator and denominator.

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

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

$$\frac{6\left(x+1\right)}{x^{4}+8x^{3}+21x^{2}+18x}$$

Use the distributive property to multiply 6 by x+1.

$$\frac{6x+6}{x^{4}+8x^{3}+21x^{2}+18x}$$