Simplify (-2/(x+2)-4/(x+1))/(-4/(x+2)-2/(x+1))

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

Step 1

Multiply the numerator and denominator of the complex fraction by $(x + 2) (x + 1)$ .

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

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

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

Cancel the common factor of $x + 2$ .

Move the leading negative in $- \frac{2}{x + 2}$ into the numerator.

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x + 1$ .

Move the leading negative in $- \frac{4}{x + 1}$ into the numerator.

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x + 2$ .

Move the leading negative in $- \frac{4}{x + 2}$ into the numerator.

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x + 1$ .

Move the leading negative in $- \frac{2}{x + 1}$ into the numerator.

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

Cancel the common factor of $(x + 1) \cdot - 2 + (x + 2) \cdot - 4$ and $(x + 1) \cdot - 4 + (x + 2) \cdot - 2$ .

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

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

Factor $2$ out of $(x + 2) \cdot - 4$ .

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

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

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

Cancel the common factors.

Factor $2$ out of $(x + 1) \cdot - 4$ .

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

Factor $2$ out of $(x + 2) \cdot - 2$ .

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 5

Simplify the numerator.

Apply the distributive property.

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

Move $- 1$ to the left of $x$ .

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

Multiply $- 1$ by $1$ .

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

Rewrite $- 1 x$ as $- x$ .

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

Apply the distributive property.

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

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

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

Multiply $2$ by $- 2$ .

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

Subtract $2 x$ from $- x$ .

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

Subtract $4$ from $- 1$ .

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

Step 6

Simplify the denominator.

Apply the distributive property.

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

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

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

Multiply $- 2$ by $1$ .

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

Apply the distributive property.

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

Move $- 1$ to the left of $x$ .

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

Multiply $2$ by $- 1$ .

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

Rewrite $- 1 x$ as $- x$ .

$\frac{- 3 x - 5}{- 2 x - 2 - x - 2}$

Subtract $x$ from $- 2 x$ .

$\frac{- 3 x - 5}{- 3 x - 2 - 2}$

Subtract $2$ from $- 2$ .

$\frac{- 3 x - 5}{- 3 x - 4}$

Step 7

Simplify terms.

Factor $- 1$ out of $- 3 x$ .

$\frac{- (3 x) - 5}{- 3 x - 4}$

Rewrite $- 5$ as $- 1 (5)$ .

$\frac{- (3 x) - 1 (5)}{- 3 x - 4}$

Factor $- 1$ out of $- (3 x) - 1 (5)$ .

$\frac{- (3 x + 5)}{- 3 x - 4}$

Rewrite $- (3 x + 5)$ as $- 1 (3 x + 5)$ .

$\frac{- 1 (3 x + 5)}{- 3 x - 4}$

Factor $- 1$ out of $- 3 x$ .

$\frac{- 1 (3 x + 5)}{- (3 x) - 4}$

Rewrite $- 4$ as $- 1 (4)$ .

$\frac{- 1 (3 x + 5)}{- (3 x) - 1 (4)}$

Factor $- 1$ out of $- (3 x) - 1 (4)$ .

$\frac{- 1 (3 x + 5)}{- (3 x + 4)}$

Rewrite $- (3 x + 4)$ as $- 1 (3 x + 4)$ .

$\frac{- 1 (3 x + 5)}{- 1 (3 x + 4)}$

Cancel the common factor.

$\frac{- 1 (3 x + 5)}{- 1 (3 x + 4)}$

Rewrite the expression.

$\frac{3 x + 5}{3 x + 4}$

Simplify (-2/(x+2)-4/(x+1))/(-4/(x+2)-2/(x+1))

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