$$\dfrac{(x-1)(x+2)(x-3)(x+3)(x+5)}{(x+2)(x-1)(x+4)(x-4)}=\dfrac{x^3+5x^2-9x-45}{x^2-16}$$
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{(x-1)(x+2)(x-3)(x+3)(x+5)}{(x+2)(x-1)(x+4)(x-4)}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{(x^2+2x-x-2)(x-3)(x+3)(x+5)}{(x^2-x+2x-2)(x+4)(x-4)} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{(x^2+x-2)(x-3)(x+3)(x+5)}{(x^2+x-2)(x+4)(x-4)} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}\frac{(x^3-3x^2+x^2-3x-2x+6)(x+3)(x+5)}{(x^3+4x^2+x^2+4x-2x-8)(x-4)} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}\frac{(x^3-2x^2-5x+6)(x+3)(x+5)}{(x^3+5x^2+2x-8)(x-4)} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle9}{\textcircled {9}} \htmlClass{explanationCircle explanationCircle10}{\textcircled {10}} \htmlClass{explanationCircle explanationCircle11}{\textcircled {11}} } }}}\frac{(x^4+x^3-11x^2-9x+18)(x+5)}{(x^3+5x^2+2x-8)(x-4)} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle12}{\textcircled {12}} \htmlClass{explanationCircle explanationCircle13}{\textcircled {13}} \htmlClass{explanationCircle explanationCircle14}{\textcircled {14}} \htmlClass{explanationCircle explanationCircle15}{\textcircled {15}} } }}}\frac{x^5+6x^4-6x^3-64x^2-27x+90}{x^4+x^3-18x^2-16x+32} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle16}{\textcircled {16}} } }}}\frac{x^3+5x^2-9x-45}{x^2-16}\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{x-1}\right) $ by each term in $ \left( x+2\right) $. $$ \left( \color{blue}{x-1}\right) \cdot \left( x+2\right) = x^2+2x-x-2 $$ |
| ② | Multiply each term of $ \left( \color{blue}{x+2}\right) $ by each term in $ \left( x-1\right) $. $$ \left( \color{blue}{x+2}\right) \cdot \left( x-1\right) = x^2-x+2x-2 $$ |
| ③ | Combine like terms: $$ x^2+ \color{blue}{2x} \color{blue}{-x} -2 = x^2+ \color{blue}{x} -2 $$ |
| ④ | Combine like terms: $$ x^2 \color{blue}{-x} + \color{blue}{2x} -2 = x^2+ \color{blue}{x} -2 $$ |
| ⑤ | Multiply each term of $ \left( \color{blue}{x^2+x-2}\right) $ by each term in $ \left( x-3\right) $. $$ \left( \color{blue}{x^2+x-2}\right) \cdot \left( x-3\right) = x^3-3x^2+x^2-3x-2x+6 $$ |
| ⑥ | Multiply each term of $ \left( \color{blue}{x^2+x-2}\right) $ by each term in $ \left( x+4\right) $. $$ \left( \color{blue}{x^2+x-2}\right) \cdot \left( x+4\right) = x^3+4x^2+x^2+4x-2x-8 $$ |
| ⑦ | Combine like terms: $$ x^3 \color{blue}{-3x^2} + \color{blue}{x^2} \color{red}{-3x} \color{red}{-2x} +6 = x^3 \color{blue}{-2x^2} \color{red}{-5x} +6 $$ |
| ⑧ | Combine like terms: $$ x^3+ \color{blue}{4x^2} + \color{blue}{x^2} + \color{red}{4x} \color{red}{-2x} -8 = x^3+ \color{blue}{5x^2} + \color{red}{2x} -8 $$ |
| ⑨ | Multiply each term of $ \left( \color{blue}{x^3-2x^2-5x+6}\right) $ by each term in $ \left( x+3\right) $. $$ \left( \color{blue}{x^3-2x^2-5x+6}\right) \cdot \left( x+3\right) = x^4+3x^3-2x^3-6x^2-5x^2-15x+6x+18 $$ |
| ⑩ | Combine like terms: $$ x^4+ \color{blue}{3x^3} \color{blue}{-2x^3} \color{red}{-6x^2} \color{red}{-5x^2} \color{green}{-15x} + \color{green}{6x} +18 = x^4+ \color{blue}{x^3} \color{red}{-11x^2} \color{green}{-9x} +18 $$ |
| ⑪ | Combine like terms: $$ x^3+ \color{blue}{4x^2} + \color{blue}{x^2} + \color{red}{4x} \color{red}{-2x} -8 = x^3+ \color{blue}{5x^2} + \color{red}{2x} -8 $$ |
| ⑫ | Multiply each term of $ \left( \color{blue}{x^4+x^3-11x^2-9x+18}\right) $ by each term in $ \left( x+5\right) $. $$ \left( \color{blue}{x^4+x^3-11x^2-9x+18}\right) \cdot \left( x+5\right) = x^5+5x^4+x^4+5x^3-11x^3-55x^2-9x^2-45x+18x+90 $$ |
| ⑬ | Combine like terms: $$ x^5+ \color{blue}{5x^4} + \color{blue}{x^4} + \color{red}{5x^3} \color{red}{-11x^3} \color{green}{-55x^2} \color{green}{-9x^2} \color{orange}{-45x} + \color{orange}{18x} +90 = \\ = x^5+ \color{blue}{6x^4} \color{red}{-6x^3} \color{green}{-64x^2} \color{orange}{-27x} +90 $$ |
| ⑭ | Multiply each term of $ \left( \color{blue}{x^3+5x^2+2x-8}\right) $ by each term in $ \left( x-4\right) $. $$ \left( \color{blue}{x^3+5x^2+2x-8}\right) \cdot \left( x-4\right) = x^4-4x^3+5x^3-20x^2+2x^2-8x-8x+32 $$ |
| ⑮ | Combine like terms: $$ x^4 \color{blue}{-4x^3} + \color{blue}{5x^3} \color{red}{-20x^2} + \color{red}{2x^2} \color{green}{-8x} \color{green}{-8x} +32 = x^4+ \color{blue}{x^3} \color{red}{-18x^2} \color{green}{-16x} +32 $$ |
| ⑯ | Simplify $ \dfrac{x^5+6x^4-6x^3-64x^2-27x+90}{x^4+x^3-18x^2-16x+32} $ to $ \dfrac{x^3+5x^2-9x-45}{x^2-16} $. Factor both the denominator and the numerator, then cancel the common factor. $\color{blue}{x^2+x-2}$. $$ \begin{aligned} \frac{x^5+6x^4-6x^3-64x^2-27x+90}{x^4+x^3-18x^2-16x+32} & =\frac{ \left( x^3+5x^2-9x-45 \right) \cdot \color{blue}{ \left( x^2+x-2 \right) }}{ \left( x^2-16 \right) \cdot \color{blue}{ \left( x^2+x-2 \right) }} = \\[1ex] &= \frac{x^3+5x^2-9x-45}{x^2-16} \end{aligned} $$ |