Simplify (2x+10)/(x^2+x-2)+2/(x+2)+x/(x-1)

$\frac{2 x + 10}{x^{2} + x - 2} + \frac{2}{x + 2} + \frac{x}{x - 1}$

Simplify each term.

Factor $2$ out of $2 x + 10$ .

Factor $2$ out of $2 x$ .

$\frac{2 (x) + 10}{x^{2} + x - 2} + \frac{2}{x + 2} + \frac{x}{x - 1}$

Factor $2$ out of $10$ .

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

Factor $2$ out of $2 x + 2 \cdot 5$ .

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

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.

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

Find the common denominator.

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

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

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

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

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

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

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

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

Simplify terms.

Combine the numerators over the common denominator.

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

Simplify each term.

Apply the distributive property.

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

Multiply $2$ by $5$ .

$\frac{2 x + 10 + 2 (x - 1) + x (x + 2)}{(x + 2) (x - 1)}$

Apply the distributive property.

$\frac{2 x + 10 + 2 x + 2 \cdot - 1 + x (x + 2)}{(x + 2) (x - 1)}$

Multiply $2$ by $- 1$ .

$\frac{2 x + 10 + 2 x - 2 + x (x + 2)}{(x + 2) (x - 1)}$

Apply the distributive property.

$\frac{2 x + 10 + 2 x - 2 + x \cdot x + x \cdot 2}{(x + 2) (x - 1)}$

Multiply $x$ by $x$ .

$\frac{2 x + 10 + 2 x - 2 + x^{2} + x \cdot 2}{(x + 2) (x - 1)}$

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

$\frac{2 x + 10 + 2 x - 2 + x^{2} + 2 x}{(x + 2) (x - 1)}$

Simplify by adding terms.

Add $2 x$ and $2 x$ .

$\frac{4 x + 10 - 2 + x^{2} + 2 x}{(x + 2) (x - 1)}$

Add $4 x$ and $2 x$ .

$\frac{6 x + 10 - 2 + x^{2}}{(x + 2) (x - 1)}$

Simplify the expression.

Subtract $2$ from $10$ .

$\frac{6 x + 8 + x^{2}}{(x + 2) (x - 1)}$

Move $8$ .

$\frac{6 x + x^{2} + 8}{(x + 2) (x - 1)}$

Reorder $6 x$ and $x^{2}$ .

$\frac{x^{2} + 6 x + 8}{(x + 2) (x - 1)}$

Factor $x^{2} + 6 x + 8$ 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 $8$ and whose sum is $6$ .

$2, 4$

Write the factored form using these integers.

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

Cancel the common factor of $x + 2$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{x + 4}{x - 1}$

Simplify (2x+10)/(x^2+x-2)+2/(x+2)+x/(x-1)

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