Simplify (x-1)/(x^2-2x-3)+5/(2x^2+2x)

$\frac{x - 1}{x^{2} - 2 x - 3} + \frac{5}{2 x^{2} + 2 x}$

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 $- 2$ .

$- 3, 1$

Write the factored form using these integers.

$\frac{x - 1}{(x - 3) (x + 1)} + \frac{5}{2 x^{2} + 2 x}$

Factor $2 x$ out of $2 x^{2} + 2 x$ .

Factor $2 x$ out of $2 x^{2}$ .

$\frac{x - 1}{(x - 3) (x + 1)} + \frac{5}{2 x (x) + 2 x}$

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

$\frac{x - 1}{(x - 3) (x + 1)} + \frac{5}{2 x (x) + 2 x (1)}$

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

$\frac{x - 1}{(x - 3) (x + 1)} + \frac{5}{2 x (x + 1)}$

Step 2

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

$\frac{x - 1}{(x - 3) (x + 1)} \cdot \frac{2 x}{2 x} + \frac{5}{2 x (x + 1)}$

Step 3

To write $\frac{5}{2 x (x + 1)}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

$\frac{x - 1}{(x - 3) (x + 1)} \cdot \frac{2 x}{2 x} + \frac{5}{2 x (x + 1)} \cdot \frac{x - 3}{x - 3}$

Step 4

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

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

$\frac{(x - 1) (2 x)}{(x - 3) (x + 1) (2 x)} + \frac{5}{2 x (x + 1)} \cdot \frac{x - 3}{x - 3}$

Multiply $\frac{5}{2 x (x + 1)}$ by $\frac{x - 3}{x - 3}$ .

$\frac{(x - 1) (2 x)}{(x - 3) (x + 1) (2 x)} + \frac{5 (x - 3)}{2 x (x + 1) (x - 3)}$

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

$\frac{(x - 1) (2 x)}{2 x (x + 1) (x - 3)} + \frac{5 (x - 3)}{2 x (x + 1) (x - 3)}$

Step 5

Combine the numerators over the common denominator.

$\frac{(x - 1) (2 x) + 5 (x - 3)}{2 x (x + 1) (x - 3)}$

Step 6

Simplify the numerator.

Apply the distributive property.

$\frac{(x \cdot 2 - 1 \cdot 2) x + 5 (x - 3)}{2 x (x + 1) (x - 3)}$

Move $2$ to the left of $x$ .

$\frac{(2 \cdot x - 1 \cdot 2) x + 5 (x - 3)}{2 x (x + 1) (x - 3)}$

Multiply $- 1$ by $2$ .

$\frac{(2 \cdot x - 2) x + 5 (x - 3)}{2 x (x + 1) (x - 3)}$

Apply the distributive property.

$\frac{2 x \cdot x - 2 x + 5 (x - 3)}{2 x (x + 1) (x - 3)}$

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

Move $x$ .

$\frac{2 (x \cdot x) - 2 x + 5 (x - 3)}{2 x (x + 1) (x - 3)}$

Multiply $x$ by $x$ .

$\frac{2 x^{2} - 2 x + 5 (x - 3)}{2 x (x + 1) (x - 3)}$

Apply the distributive property.

$\frac{2 x^{2} - 2 x + 5 x + 5 \cdot - 3}{2 x (x + 1) (x - 3)}$

Multiply $5$ by $- 3$ .

$\frac{2 x^{2} - 2 x + 5 x - 15}{2 x (x + 1) (x - 3)}$

Add $- 2 x$ and $5 x$ .

$\frac{2 x^{2} + 3 x - 15}{2 x (x + 1) (x - 3)}$

Simplify (x-1)/(x^2-2x-3)+5/(2x^2+2x)

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