Solve left(x+1right)left(1+x+2/x-2+x+3/x-3left(1+x+2/x-2right)right)

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

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x-2}{x-2}.

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

Since \frac{x-2}{x-2} and \frac{x+2}{x-2} have the same denominator, add them by adding their numerators.

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

Combine like terms in x-2+x+2.

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x-2}{x-2}.

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

Since \frac{x-2}{x-2} and \frac{x+2}{x-2} have the same denominator, add them by adding their numerators.

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

Combine like terms in x-2+x+2.

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

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

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

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

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

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

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

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

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

Express \left(x+1\right)\times \frac{4x^{2}}{\left(x-3\right)\left(x-2\right)} as a single fraction.

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

Use the distributive property to multiply x+1 by 4.

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

Use the distributive property to multiply 4x+4 by x^{2}.

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

Apply the distributive property by multiplying each term of x-3 by each term of x-2.

$$\frac{4x^{3}+4x^{2}}{x^{2}-2x-3x+6}$$

Combine -2x and -3x to get -5x.

$$\frac{4x^{3}+4x^{2}}{x^{2}-5x+6}$$

Expand

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x-2}{x-2}.

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

Since \frac{x-2}{x-2} and \frac{x+2}{x-2} have the same denominator, add them by adding their numerators.

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

Combine like terms in x-2+x+2.

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x-2}{x-2}.

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

Since \frac{x-2}{x-2} and \frac{x+2}{x-2} have the same denominator, add them by adding their numerators.

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

Combine like terms in x-2+x+2.

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

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

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

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

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

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

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

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

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

Express \left(x+1\right)\times \frac{4x^{2}}{\left(x-3\right)\left(x-2\right)} as a single fraction.

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

Use the distributive property to multiply x+1 by 4.

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

Use the distributive property to multiply 4x+4 by x^{2}.

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

Apply the distributive property by multiplying each term of x-3 by each term of x-2.

$$\frac{4x^{3}+4x^{2}}{x^{2}-2x-3x+6}$$

Combine -2x and -3x to get -5x.

$$\frac{4x^{3}+4x^{2}}{x^{2}-5x+6}$$