Simplify (3+n-2n^2)/(1+n)

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

Step 1

Factor by grouping.

Reorder terms.

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

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 $a \cdot c = - 2 \cdot 3 = - 6$ and whose sum is $b = 1$ .

Multiply by $1$ .

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

Rewrite $1$ as $- 2$ plus $3$

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

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

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

Step 2

Cancel the common factor of $- n - 1$ and $1 + n$ .

Factor $- 1$ out of $- n$ .

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

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

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

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

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

Rewrite $- (n + 1)$ as $- 1 (n + 1)$ .

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

Reorder terms.

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

Cancel the common factor.

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

Divide $- 1 (2 n - 3)$ by $1$ .

$- 1 (2 n - 3)$

Step 3

Rewrite $- 1 (2 n - 3)$ as $- (2 n - 3)$ .

$- (2 n - 3)$

Step 4

Apply the distributive property.

$- (2 n) - - 3$

Step 5

Multiply.

Multiply $2$ by $- 1$ .

$- 2 n - - 3$

Multiply $- 1$ by $- 3$ .

$- 2 n + 3$

Simplify (3+n-2n^2)/(1+n)

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