Simplify (x-2)/(x+1)+(5x-30)/(x^2+7x+6)

$\frac{(x - 2)}{(x + 1)} + \frac{(5 x - 30)}{(x^{2} + 7 x + 6)}$

Step 1

Simplify each term.

Factor $5$ out of $5 x - 30$ .

Factor $5$ out of $5 x$ .

$\frac{x - 2}{x + 1} + \frac{5 (x) - 30}{x^{2} + 7 x + 6}$

Factor $5$ out of $- 30$ .

$\frac{x - 2}{x + 1} + \frac{5 x + 5 \cdot - 6}{x^{2} + 7 x + 6}$

Factor $5$ out of $5 x + 5 \cdot - 6$ .

$\frac{x - 2}{x + 1} + \frac{5 (x - 6)}{x^{2} + 7 x + 6}$

Factor $x^{2} + 7 x + 6$ 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 $6$ and whose sum is $7$ .

$1, 6$

Write the factored form using these integers.

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

Step 2

To write $\frac{x - 2}{x + 1}$ as a fraction with a common denominator, multiply by $\frac{x + 6}{x + 6}$ .

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

Step 3

Simplify terms.

Multiply $\frac{x - 2}{x + 1}$ by $\frac{x + 6}{x + 6}$ .

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

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

Expand $(x - 2) (x + 6)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $6$ to the left of $x$ .

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

Multiply $- 2$ by $6$ .

$\frac{x^{2} + 6 x - 2 x - 12 + 5 (x - 6)}{(x + 1) (x + 6)}$

Subtract $2 x$ from $6 x$ .

$\frac{x^{2} + 4 x - 12 + 5 (x - 6)}{(x + 1) (x + 6)}$

Apply the distributive property.

$\frac{x^{2} + 4 x - 12 + 5 x + 5 \cdot - 6}{(x + 1) (x + 6)}$

Multiply $5$ by $- 6$ .

$\frac{x^{2} + 4 x - 12 + 5 x - 30}{(x + 1) (x + 6)}$

Add $4 x$ and $5 x$ .

$\frac{x^{2} + 9 x - 12 - 30}{(x + 1) (x + 6)}$

Subtract $30$ from $- 12$ .

$\frac{x^{2} + 9 x - 42}{(x + 1) (x + 6)}$

Simplify (x-2)/(x+1)+(5x-30)/(x^2+7x+6)

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