Simplify (2x+3)/(x+1)+x/(x-1)

$\frac{2 x + 3}{x + 1} + \frac{x}{x - 1}$

To write $\frac{2 x + 3}{x + 1}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

$\frac{2 x + 3}{x + 1} \cdot \frac{x - 1}{x - 1} + \frac{x}{x - 1}$

To write $\frac{x}{x - 1}$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

$\frac{2 x + 3}{x + 1} \cdot \frac{x - 1}{x - 1} + \frac{x}{x - 1} \cdot \frac{x + 1}{x + 1}$

Write each expression with a common denominator of $(x + 1) (x - 1)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{2 x + 3}{x + 1}$ by $\frac{x - 1}{x - 1}$ .

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

Multiply $\frac{x}{x - 1}$ by $\frac{x + 1}{x + 1}$ .

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

Reorder the factors of $(x - 1) (x + 1)$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Expand $(2 x + 3) (x - 1)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 1$ by $2$ .

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

Multiply $3$ by $- 1$ .

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

Add $- 2 x$ and $3 x$ .

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

Apply the distributive property.

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

Multiply $x$ by $x$ .

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

Multiply $x$ by $1$ .

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

Add $2 x^{2}$ and $x^{2}$ .

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

Add $x$ and $x$ .

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