Simplify (6n^2+8n+2)/(2n^2-2)

$\frac{6 n^{2} + 8 n + 2}{2 n^{2} - 2}$

Step 1

Cancel the common factor of $6 n^{2} + 8 n + 2$ and $2 n^{2} - 2$ .

Factor $2$ out of $6 n^{2}$ .

$\frac{2 (3 n^{2}) + 8 n + 2}{2 n^{2} - 2}$

Factor $2$ out of $8 n$ .

$\frac{2 (3 n^{2}) + 2 (4 n) + 2}{2 n^{2} - 2}$

Factor $2$ out of $2 (3 n^{2}) + 2 (4 n)$ .

$\frac{2 (3 n^{2} + 4 n) + 2}{2 n^{2} - 2}$

Factor $2$ out of $2$ .

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

Factor $2$ out of $2 (3 n^{2} + 4 n) + 2 (1)$ .

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

Cancel the common factors.

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

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

Factor $2$ out of $- 2$ .

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

Factor $2$ out of $2 (n^{2}) + 2 (- 1)$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{3 n^{2} + 4 n + 1}{n^{2} - 1}$

Step 2

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot 1 = 3$ and whose sum is $b = 4$ .

Factor $4$ out of $4 n$ .

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

Rewrite $4$ as $1$ plus $3$

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

Apply the distributive property.

$\frac{3 n^{2} + 1 n + 3 n + 1}{n^{2} - 1}$

Multiply $n$ by $1$ .

$\frac{3 n^{2} + n + 3 n + 1}{n^{2} - 1}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

Factor out the greatest common factor (GCF) from each group.

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

Factor the polynomial by factoring out the greatest common factor, $3 n + 1$ .

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

Step 3

Simplify the denominator.

Rewrite $1$ as $1^{2}$ .

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

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = n$ and $b = 1$ .

$\frac{(3 n + 1) (n + 1)}{(n + 1) (n - 1)}$

Step 4

Cancel the common factor of $n + 1$ .

Cancel the common factor.

$\frac{(3 n + 1) (n + 1)}{(n + 1) (n - 1)}$

Rewrite the expression.

$\frac{3 n + 1}{n - 1}$

Simplify (6n^2+8n+2)/(2n^2-2)

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