Simplify ((4a)/(2-a)-a)÷((a+2)/(a-2))

$(\frac{4 a}{2 - a} - a) \div \frac{a + 2}{a - 2}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$(\frac{4 a}{2 - a} - a) \frac{a - 2}{a + 2}$

Step 2

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

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

Step 3

Simplify terms.

Combine $- a$ and $\frac{2 - a}{2 - a}$ .

$(\frac{4 a}{2 - a} + \frac{- a (2 - a)}{2 - a}) \frac{a - 2}{a + 2}$

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

Factor $a$ out of $4 a - a (2 - a)$ .

Factor $a$ out of $4 a$ .

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

Factor $a$ out of $- a (2 - a)$ .

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

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

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

Apply the distributive property.

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

Multiply $- 1$ by $2$ .

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

Multiply $- - a$ .

Multiply $- 1$ by $- 1$ .

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

Multiply $a$ by $1$ .

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

Subtract $2$ from $4$ .

$\frac{a (2 + a)}{2 - a} \cdot \frac{a - 2}{a + 2}$

Step 5

Simplify terms.

Multiply $\frac{a (2 + a)}{2 - a}$ by $\frac{a - 2}{a + 2}$ .

$\frac{a (2 + a) (a - 2)}{(2 - a) (a + 2)}$

Cancel the common factor of $2 + a$ and $a + 2$ .

Reorder terms.

$\frac{a (a + 2) (a - 2)}{(2 - a) (a + 2)}$

Cancel the common factor.

$\frac{a (a + 2) (a - 2)}{(2 - a) (a + 2)}$

Rewrite the expression.

$\frac{a (a - 2)}{2 - a}$

Cancel the common factor of $a - 2$ and $2 - a$ .

Factor $- 1$ out of $a$ .

$\frac{a (- 1 (- a) - 2)}{2 - a}$

Rewrite $- 2$ as $- 1 (2)$ .

$\frac{a (- 1 (- a) - 1 (2))}{2 - a}$

Factor $- 1$ out of $- 1 (- a) - 1 (2)$ .

$\frac{a (- 1 (- a + 2))}{2 - a}$

Reorder terms.

$\frac{a (- 1 (- a + 2))}{- a + 2}$

Cancel the common factor.

$\frac{a (- 1 (- a + 2))}{- a + 2}$

Divide $a \cdot (- 1)$ by $1$ .

$a \cdot (- 1)$

Simplify the expression.

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

$- 1 a$

Rewrite $- 1 a$ as $- a$ .

$- a$

Simplify ((4a)/(2-a)-a)÷((a+2)/(a-2))

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