$$\dfrac{\dfrac{\dfrac{m^2+4m-21}{m^2+8m+15}}{m^2-9}}{m^2+12m+35}=\dfrac{1}{m^4+16m^3+94m^2+240m+225}$$
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\frac{\frac{m^2+4m-21}{m^2+8m+15}}{m^2-9}}{m^2+12m+35}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\frac{m+7}{m^3+11m^2+39m+45}}{m^2+12m+35} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{1}{m^4+16m^3+94m^2+240m+225}\end{aligned} $$ | |
| ① | Divide $ \dfrac{m^2+4m-21}{m^2+8m+15} $ by $ m^2-9 $ to get $ \dfrac{m+7}{m^3+11m^2+39m+45} $. Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction. Step 2: Factor numerators and denominators. Step 3: Cancel common factors. Step 4: Multiply numerators and denominators. Step 5: Simplify numerator and denominator. $$ \begin{aligned} \frac{ \frac{m^2+4m-21}{m^2+8m+15} }{m^2-9} & \xlongequal{\text{Step 1}} \frac{m^2+4m-21}{m^2+8m+15} \cdot \frac{\color{blue}{1}}{\color{blue}{m^2-9}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( m+7 \right) \cdot \color{blue}{ \left( m-3 \right) } }{ m^2+8m+15 } \cdot \frac{ 1 }{ \left( m+3 \right) \cdot \color{blue}{ \left( m-3 \right) } } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ m+7 }{ m^2+8m+15 } \cdot \frac{ 1 }{ m+3 } \xlongequal{\text{Step 4}} \frac{ \left( m+7 \right) \cdot 1 }{ \left( m^2+8m+15 \right) \cdot \left( m+3 \right) } = \\[1ex] & \xlongequal{\text{Step 5}} \frac{ m+7 }{ m^3+3m^2+8m^2+24m+15m+45 } = \frac{m+7}{m^3+11m^2+39m+45} \end{aligned} $$ |
| ② | Divide $ \dfrac{m+7}{m^3+11m^2+39m+45} $ by $ m^2+12m+35 $ to get $ \dfrac{1}{m^4+16m^3+94m^2+240m+225} $. Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction. Step 2: Factor numerators and denominators. Step 3: Cancel common factors. Step 4: Multiply numerators and denominators. Step 5: Simplify numerator and denominator. $$ \begin{aligned} \frac{ \frac{m+7}{m^3+11m^2+39m+45} }{m^2+12m+35} & \xlongequal{\text{Step 1}} \frac{m+7}{m^3+11m^2+39m+45} \cdot \frac{\color{blue}{1}}{\color{blue}{m^2+12m+35}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ 1 \cdot \color{blue}{ \left( m+7 \right) } }{ m^3+11m^2+39m+45 } \cdot \frac{ 1 }{ \left( m+5 \right) \cdot \color{blue}{ \left( m+7 \right) } } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 1 }{ m^3+11m^2+39m+45 } \cdot \frac{ 1 }{ m+5 } \xlongequal{\text{Step 4}} \frac{ 1 \cdot 1 }{ \left( m^3+11m^2+39m+45 \right) \cdot \left( m+5 \right) } = \\[1ex] & \xlongequal{\text{Step 5}} \frac{ 1 }{ m^4+5m^3+11m^3+55m^2+39m^2+195m+45m+225 } = \frac{1}{m^4+16m^3+94m^2+240m+225} \end{aligned} $$ |