Simplify (-3n+12)/(84n^2+76n+16)

$\frac{- 3 n + 12}{84 n^{2} + 76 n + 16}$

Factor $3$ out of $- 3 n + 12$ .

Factor $3$ out of $- 3 n$ .

$\frac{3 (- n) + 12}{84 n^{2} + 76 n + 16}$

Factor $3$ out of $12$ .

$\frac{3 (- n) + 3 (4)}{84 n^{2} + 76 n + 16}$

Factor $3$ out of $3 (- n) + 3 (4)$ .

$\frac{3 (- n + 4)}{84 n^{2} + 76 n + 16}$

Simplify the denominator.

Factor $4$ out of $84 n^{2} + 76 n + 16$ .

Factor $4$ out of $84 n^{2}$ .

$\frac{3 (- n + 4)}{4 (21 n^{2}) + 76 n + 16}$

Factor $4$ out of $76 n$ .

$\frac{3 (- n + 4)}{4 (21 n^{2}) + 4 (19 n) + 16}$

Factor $4$ out of $16$ .

$\frac{3 (- n + 4)}{4 (21 n^{2}) + 4 (19 n) + 4 (4)}$

Factor $4$ out of $4 (21 n^{2}) + 4 (19 n)$ .

$\frac{3 (- n + 4)}{4 (21 n^{2} + 19 n) + 4 (4)}$

Factor $4$ out of $4 (21 n^{2} + 19 n) + 4 (4)$ .

$\frac{3 (- n + 4)}{4 (21 n^{2} + 19 n + 4)}$

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 $a \cdot c = 21 \cdot 4 = 84$ and whose sum is $b = 19$ .

Factor $19$ out of $19 n$ .

$\frac{3 (- n + 4)}{4 (21 n^{2} + 19 (n) + 4)}$

Rewrite $19$ as $7$ plus $12$

$\frac{3 (- n + 4)}{4 (21 n^{2} + (7 + 12) n + 4)}$

Apply the distributive property.

$\frac{3 (- n + 4)}{4 (21 n^{2} + 7 n + 12 n + 4)}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{3 (- n + 4)}{4 ((21 n^{2} + 7 n) + 12 n + 4)}$

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

$\frac{3 (- n + 4)}{4 (7 n (3 n + 1) + 4 (3 n + 1))}$

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

$\frac{3 (- n + 4)}{4 ((3 n + 1) (7 n + 4))}$

$\frac{3 (- n + 4)}{4 (3 n + 1) (7 n + 4)}$

Simplify with factoring out.

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

$\frac{3 (- (n) + 4)}{4 (3 n + 1) (7 n + 4)}$

Rewrite $4$ as $- 1 (- 4)$ .

$\frac{3 (- (n) - 1 (- 4))}{4 (3 n + 1) (7 n + 4)}$

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

$\frac{3 (- (n - 4))}{4 (3 n + 1) (7 n + 4)}$

Simplify the expression.

Rewrite $- (n - 4)$ as $- 1 (n - 4)$ .

$\frac{3 (- 1 (n - 4))}{4 (3 n + 1) (7 n + 4)}$

Move the negative in front of the fraction.

$- \frac{3 (n - 4)}{4 (3 n + 1) (7 n + 4)}$

Simplify (-3n+12)/(84n^2+76n+16)

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