Simplify (2m^2+12m)/(4m^2-36)+18/(4m^2-36)

$\frac{2 m^{2} + 12 m}{4 m^{2} - 36} + \frac{18}{4 m^{2} - 36}$

Simplify terms.

Combine the numerators over the common denominator.

$\frac{2 m^{2} + 12 m + 18}{4 m^{2} - 36}$

Cancel the common factor of $2 m^{2} + 12 m + 18$ and $4 m^{2} - 36$ .

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

$\frac{2 (m^{2}) + 12 m + 18}{4 m^{2} - 36}$

Factor $2$ out of $12 m$ .

$\frac{2 (m^{2}) + 2 (6 m) + 18}{4 m^{2} - 36}$

Factor $2$ out of $2 (m^{2}) + 2 (6 m)$ .

$\frac{2 (m^{2} + 6 m) + 18}{4 m^{2} - 36}$

Factor $2$ out of $18$ .

$\frac{2 (m^{2} + 6 m) + 2 \cdot 9}{4 m^{2} - 36}$

Factor $2$ out of $2 (m^{2} + 6 m) + 2 \cdot 9$ .

$\frac{2 (m^{2} + 6 m + 9)}{4 m^{2} - 36}$

Cancel the common factors.

Factor $2$ out of $4 m^{2}$ .

$\frac{2 (m^{2} + 6 m + 9)}{2 (2 m^{2}) - 36}$

Factor $2$ out of $- 36$ .

$\frac{2 (m^{2} + 6 m + 9)}{2 (2 m^{2}) + 2 (- 18)}$

Factor $2$ out of $2 (2 m^{2}) + 2 (- 18)$ .

$\frac{2 (m^{2} + 6 m + 9)}{2 (2 m^{2} - 18)}$

Cancel the common factor.

$\frac{2 (m^{2} + 6 m + 9)}{2 (2 m^{2} - 18)}$

Rewrite the expression.

$\frac{m^{2} + 6 m + 9}{2 m^{2} - 18}$

Factor using the perfect square rule.

Rewrite $9$ as $3^{2}$ .

$\frac{m^{2} + 6 m + 3^{2}}{2 m^{2} - 18}$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$6 m = 2 \cdot m \cdot 3$

Rewrite the polynomial.

$\frac{m^{2} + 2 \cdot m \cdot 3 + 3^{2}}{2 m^{2} - 18}$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = m$ and $b = 3$ .

$\frac{{(m + 3)}^{2}}{2 m^{2} - 18}$

Simplify the denominator.

Factor $2$ out of $2 m^{2} - 18$ .

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

$\frac{{(m + 3)}^{2}}{2 (m^{2}) - 18}$

Factor $2$ out of $- 18$ .

$\frac{{(m + 3)}^{2}}{2 m^{2} + 2 \cdot - 9}$

Factor $2$ out of $2 m^{2} + 2 \cdot - 9$ .

$\frac{{(m + 3)}^{2}}{2 (m^{2} - 9)}$

Rewrite $9$ as $3^{2}$ .

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

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = m$ and $b = 3$ .

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

Cancel the common factor of ${(m + 3)}^{2}$ and $m + 3$ .

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

$\frac{(m + 3) (m + 3)}{2 (m + 3) (m - 3)}$

Cancel the common factors.

Factor $m + 3$ out of $2 (m + 3) (m - 3)$ .

$\frac{(m + 3) (m + 3)}{(m + 3) (2 (m - 3))}$

Cancel the common factor.

$\frac{(m + 3) (m + 3)}{(m + 3) (2 (m - 3))}$

Rewrite the expression.

$\frac{m + 3}{2 (m - 3)}$

Simplify (2m^2+12m)/(4m^2-36)+18/(4m^2-36)

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