Simplify 6/(n+1)-(6n)/(n+4)

$\frac{6}{n + 1} - \frac{6 n}{n + 4}$

Step 1

To write $\frac{6}{n + 1}$ as a fraction with a common denominator, multiply by $\frac{n + 4}{n + 4}$ .

$\frac{6}{n + 1} \cdot \frac{n + 4}{n + 4} - \frac{6 n}{n + 4}$

Step 2

To write $- \frac{6 n}{n + 4}$ as a fraction with a common denominator, multiply by $\frac{n + 1}{n + 1}$ .

$\frac{6}{n + 1} \cdot \frac{n + 4}{n + 4} - \frac{6 n}{n + 4} \cdot \frac{n + 1}{n + 1}$

Step 3

Write each expression with a common denominator of $(n + 1) (n + 4)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{6}{n + 1}$ by $\frac{n + 4}{n + 4}$ .

$\frac{6 (n + 4)}{(n + 1) (n + 4)} - \frac{6 n}{n + 4} \cdot \frac{n + 1}{n + 1}$

Multiply $\frac{6 n}{n + 4}$ by $\frac{n + 1}{n + 1}$ .

$\frac{6 (n + 4)}{(n + 1) (n + 4)} - \frac{6 n (n + 1)}{(n + 4) (n + 1)}$

Reorder the factors of $(n + 4) (n + 1)$ .

$\frac{6 (n + 4)}{(n + 1) (n + 4)} - \frac{6 n (n + 1)}{(n + 1) (n + 4)}$

Step 4

Combine the numerators over the common denominator.

$\frac{6 (n + 4) - 6 n (n + 1)}{(n + 1) (n + 4)}$

Step 5

Simplify the numerator.

Factor $6$ out of $6 (n + 4) - 6 n (n + 1)$ .

Factor $6$ out of $- 6 n (n + 1)$ .

$\frac{6 (n + 4) + 6 (- n (n + 1))}{(n + 1) (n + 4)}$

Factor $6$ out of $6 (n + 4) + 6 (- n (n + 1))$ .

$\frac{6 (n + 4 - n (n + 1))}{(n + 1) (n + 4)}$

Apply the distributive property.

$\frac{6 (n + 4 - n \cdot n - n \cdot 1)}{(n + 1) (n + 4)}$

Multiply $n$ by $n$ by adding the exponents.

Move $n$ .

$\frac{6 (n + 4 - (n \cdot n) - n \cdot 1)}{(n + 1) (n + 4)}$

Multiply $n$ by $n$ .

$\frac{6 (n + 4 - n^{2} - n \cdot 1)}{(n + 1) (n + 4)}$

Multiply $- 1$ by $1$ .

$\frac{6 (n + 4 - n^{2} - n)}{(n + 1) (n + 4)}$

Subtract $n$ from $n$ .

$\frac{6 (4 - n^{2} + 0)}{(n + 1) (n + 4)}$

Add $4 - n^{2}$ and $0$ .

$\frac{6 (4 - n^{2})}{(n + 1) (n + 4)}$

Rewrite $4 - n^{2}$ in a factored form.

Rewrite $4$ as $2^{2}$ .

$\frac{6 (2^{2} - n^{2})}{(n + 1) (n + 4)}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2$ and $b = n$ .

$\frac{6 ((2 + n) (2 - n))}{(n + 1) (n + 4)}$

$\frac{6 (2 + n) (2 - n)}{(n + 1) (n + 4)}$