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

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

Step 1

Simplify each term.

Factor $x^{2} + 2 x - 3$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 3$ and whose sum is $2$ .

$- 1, 3$

Write the factored form using these integers.

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

Factor $2$ out of $2 x - 2$ .

Factor $2$ out of $2 x$ .

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

Factor $2$ out of $- 2$ .

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

Factor $2$ out of $2 (x) + 2 (- 1)$ .

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

Step 2

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

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

Step 3

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

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

Step 4

Write each expression with a common denominator of $(x - 1) (x + 3) \cdot 2$ , by multiplying each by an appropriate factor of $1$ .

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

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

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

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

Reorder the factors of $(x - 1) (x + 3) \cdot 2$ .

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

Reorder the factors of $2 (x - 1) (x + 3)$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

Multiply $2$ by $2$ .

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $3$ to the left of $x$ .

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

Multiply $x$ by $1$ .

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

Multiply $3$ by $1$ .

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

Add $3 x$ and $x$ .

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

Add $4 x$ and $4 x$ .

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

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

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