Simplify (12+r-r^2)/(r^3+3r^2)

$\frac{12 + r - r^{2}}{r^{3} + 3 r^{2}}$

Factor by grouping.

Reorder terms.

$\frac{- r^{2} + r + 12}{r^{3} + 3 r^{2}}$

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 $a \cdot c = - 1 \cdot 12 = - 12$ and whose sum is $b = 1$ .

Multiply by $1$ .

$\frac{- r^{2} + 1 r + 12}{r^{3} + 3 r^{2}}$

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

$\frac{- r^{2} + (- 3 + 4) r + 12}{r^{3} + 3 r^{2}}$

Apply the distributive property.

$\frac{- r^{2} - 3 r + 4 r + 12}{r^{3} + 3 r^{2}}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(- r^{2} - 3 r) + 4 r + 12}{r^{3} + 3 r^{2}}$

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

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

Factor the polynomial by factoring out the greatest common factor, $- r - 3$ .

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

Factor $r^{2}$ out of $r^{3} + 3 r^{2}$ .

Factor $r^{2}$ out of $r^{3}$ .

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

Factor $r^{2}$ out of $3 r^{2}$ .

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

Factor $r^{2}$ out of $r^{2} r + r^{2} \cdot 3$ .

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

Cancel the common factor of $- r - 3$ and $r + 3$ .

Factor $- 1$ out of $- r$ .

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

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

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

Factor $- 1$ out of $- (r) - 1 (3)$ .

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

Rewrite $- (r + 3)$ as $- 1 (r + 3)$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{- 1 (r - 4)}{r^{2}}$

Move the negative in front of the fraction.

$- \frac{r - 4}{r^{2}}$

Simplify (12+r-r^2)/(r^3+3r^2)

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