Simplify (2x)/(x^2+x-2)-5/(3x+6)

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

Simplify each term.

Factor $x^{2} + x - 2$ 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 $- 2$ and whose sum is $1$ .

$- 1, 2$

Write the factored form using these integers.

$\frac{2 x}{(x - 1) (x + 2)} - \frac{5}{3 x + 6}$

Factor $3$ out of $3 x + 6$ .

Factor $3$ out of $3 x$ .

$\frac{2 x}{(x - 1) (x + 2)} - \frac{5}{3 (x) + 6}$

Factor $3$ out of $6$ .

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

Factor $3$ out of $3 x + 3 \cdot 2$ .

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

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

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

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

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $3$ by $2$ .

$\frac{6 x - 5 (x - 1)}{3 (x + 2) (x - 1)}$

Apply the distributive property.

$\frac{6 x - 5 x - 5 \cdot - 1}{3 (x + 2) (x - 1)}$

Multiply $- 5$ by $- 1$ .

$\frac{6 x - 5 x + 5}{3 (x + 2) (x - 1)}$

Subtract $5 x$ from $6 x$ .

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

Simplify (2x)/(x^2+x-2)-5/(3x+6)

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