Solve f(x)=(x^2+4x+3/x^2+2x-3-x^2+2x+1/x^2+3x+2)*x+2/x+1

$$f ( x ) = ( \frac { x ^ { 2 } + 4 x + 3 } { x ^ { 2 } + 2 x - 3 } - \frac { x ^ { 2 } + 2 x + 1 } { x ^ { 2 } + 3 x + 2 } ) \cdot \frac { x + 2 } { x + 1 }$$

Evaluate

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{x^{2}+4x+3}{x^{2}+2x-3}.

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

Cancel out x+3 in both numerator and denominator.

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

Factor the expressions that are not already factored in \frac{x^{2}+2x+1}{x^{2}+3x+2}.

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

Cancel out x+1 in both numerator and denominator.

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

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

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

Since \frac{\left(x+1\right)\left(x+2\right)}{\left(x-1\right)\left(x+2\right)} and \frac{\left(x+1\right)\left(x-1\right)}{\left(x-1\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x+1\right)\left(x+2\right)-\left(x+1\right)\left(x-1\right).

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

Combine like terms in x^{2}+2x+x+2-x^{2}+x-x+1.

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

Multiply \frac{3x+3}{\left(x-1\right)\left(x+2\right)} times \frac{x+2}{x+1} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out x+2 in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out x+1 in both numerator and denominator.

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

Expand

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{x^{2}+4x+3}{x^{2}+2x-3}.

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

Cancel out x+3 in both numerator and denominator.

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

Factor the expressions that are not already factored in \frac{x^{2}+2x+1}{x^{2}+3x+2}.

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

Cancel out x+1 in both numerator and denominator.

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

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

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

Since \frac{\left(x+1\right)\left(x+2\right)}{\left(x-1\right)\left(x+2\right)} and \frac{\left(x+1\right)\left(x-1\right)}{\left(x-1\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x+1\right)\left(x+2\right)-\left(x+1\right)\left(x-1\right).

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

Combine like terms in x^{2}+2x+x+2-x^{2}+x-x+1.

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

Multiply \frac{3x+3}{\left(x-1\right)\left(x+2\right)} times \frac{x+2}{x+1} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out x+2 in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out x+1 in both numerator and denominator.

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