Question

Divide Rational Expressions:$\frac{6m^2\hspace{0.17em}-\hspace{0.17em}11m\hspace{0.17em}-\hspace{0.17em}2}{9\hspace{0.17em}-\hspace{0.17em}{m}^2}÷\frac{6m^2\hspace{0.17em}+\hspace{0.17em}25m\hspace{0.17em}+\hspace{0.17em}4}{m^2\hspace{0.17em}-\hspace{0.17em}6m\hspace{0.17em}+\hspace{0.17em}9}$

Short answer

The required answer is$-\frac{\left(m-2\right)\left(m-3\right)}{\left(3+m\right)\left(m+4\right)}$.

Step-by-step solution

Step 1. Given information.

The given expression is$\frac{6m^2\hspace{0.17em}-\hspace{0.17em}11m\hspace{0.17em}-\hspace{0.17em}2}{9\hspace{0.17em}-\hspace{0.17em}{m}^2}÷\frac{6m^2\hspace{0.17em}+\hspace{0.17em}25m\hspace{0.17em}+\hspace{0.17em}4}{m^2\hspace{0.17em}-\hspace{0.17em}6m\hspace{0.17em}+\hspace{0.17em}9}$

Step 2.  Simplify the expression. 

$$\begin{gathered} \frac{6m^2\hspace{0.17em}-\hspace{0.17em}11m\hspace{0.17em}-\hspace{0.17em}2}{9\hspace{0.17em}-\hspace{0.17em}{m}^2}÷\frac{6m^2\hspace{0.17em}+\hspace{0.17em}25m\hspace{0.17em}+\hspace{0.17em}4}{m^2\hspace{0.17em}-\hspace{0.17em}6m\hspace{0.17em}+\hspace{0.17em}9} \\ =\frac{\frac{6m^2-11m-2}{9-m^2}}{\frac{6m^2+25m+4}{m^2-6m+9}} \end{gathered}$$

Apply the fraction rule.

$$\begin{gathered} \frac{\left(6m^2-11m-2\right)\left(m^2-6m+9\right)}{\left(9-m^2\right)\left(6m^2+25m+4\right)} \\ =\frac{\left(6m+1\right)\left(m-2\right)\left(m^2-6m+9\right)}{\left(9-m^2\right)\left(6m+1\right)\left(m+4\right)} \\ =\frac{\left(m-2\right)\left(m^2-6m+9\right)}{\left(9-m^2\right)\left(m+4\right)} \\ =\frac{\left(m-2\right){\left(m-3\right)}^2}{\left(9-m^2\right)\left(m+4\right)} \\ =\frac{\left(m-2\right){\left(m-3\right)}^2}{\left(3+m\right)\left(3-m\right)\left(m+4\right)} \\ =\frac{\left(m-2\right){\left(m-3\right)}^2}{-\left(3+m\right)\left(m-3\right)\left(m+4\right)} \\ =\frac{\left(m-2\right)\left(m-3\right)\left(m-3\right)}{-\left(3+m\right)\left(m-3\right)\left(m+4\right)} \\ =\frac{\left(m-2\right)\left(m-3\right)}{-\left(3+m\right)\left(m+4\right)} \\ =-\frac{\left(m-2\right)\left(m-3\right)}{\left(3+m\right)\left(m+4\right)} \end{gathered}$$