Simplify (((a-4)(a+3))/(a^2-a))÷((2(a-4))/((a+3)(a-1)))

$\frac{(a - 4) (a + 3)}{a^{2} - a} \div \frac{2 (a - 4)}{(a + 3) (a - 1)}$

Step 1

To divide by a fraction, multiply by its reciprocal.

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

Step 2

Cancel the common factor of $a - 4$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{a + 3}{a^{2} - a} \cdot \frac{(a + 3) (a - 1)}{2}$

Step 3

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

$\frac{(a + 3) ((a + 3) (a - 1))}{(a^{2} - a) \cdot 2}$

Step 4

Raise $a + 3$ to the power of $1$ .

$\frac{{(a + 3)}^{1} (a + 3) (a - 1)}{(a^{2} - a) \cdot 2}$

Step 5

Raise $a + 3$ to the power of $1$ .

$\frac{{(a + 3)}^{1} {(a + 3)}^{1} (a - 1)}{(a^{2} - a) \cdot 2}$

Step 6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{{(a + 3)}^{1 + 1} (a - 1)}{(a^{2} - a) \cdot 2}$

Step 7

Add $1$ and $1$ .

$\frac{{(a + 3)}^{2} (a - 1)}{(a^{2} - a) \cdot 2}$

Step 8

Factor $a$ out of $a^{2} - a$ .

Factor $a$ out of $a^{2}$ .

$\frac{{(a + 3)}^{2} (a - 1)}{(a \cdot a - a) \cdot 2}$

Factor $a$ out of $- a$ .

$\frac{{(a + 3)}^{2} (a - 1)}{(a \cdot a + a \cdot - 1) \cdot 2}$

Factor $a$ out of $a \cdot a + a \cdot - 1$ .

$\frac{{(a + 3)}^{2} (a - 1)}{a (a - 1) \cdot 2}$

Step 9

Cancel the common factor of $a - 1$ .

Cancel the common factor.

$\frac{{(a + 3)}^{2} (a - 1)}{a (a - 1) \cdot 2}$

Rewrite the expression.

$\frac{{(a + 3)}^{2}}{a \cdot 2}$

Step 10

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

$\frac{{(a + 3)}^{2}}{2 a}$