$$-\dfrac{1}{36}(x+4)(x+1)(x-1)(x-3)^2=\dfrac{x^5-2x^4-16x^3+38x^2+15x-36}{36}$$
Tap the blue circles to see an explanation.
| $$ \begin{aligned}-\frac{1}{36}(x+4)(x+1)(x-1)(x-3)^2& \xlongequal{ }-\frac{1}{36}(x+4)(x+1)(x-1)(x^2-6x+9) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{x+4}{36}(x+1)(x-1)(x^2-6x+9) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{x^2+5x+4}{36}(x-1)(x^2-6x+9) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{x^3+4x^2-x-4}{36}(x^2-6x+9) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{x^5-2x^4-16x^3+38x^2+15x-36}{36}\end{aligned} $$ | |
| ① | Multiply $ \dfrac{1}{36} $ by $ x+4 $ to get $ \dfrac{ x+4 }{ 36 } $. Step 1: Write $ x+4 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{1}{36} \cdot x+4 & \xlongequal{\text{Step 1}} \frac{1}{36} \cdot \frac{x+4}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 1 \cdot \left( x+4 \right) }{ 36 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ x+4 }{ 36 } \end{aligned} $$ |
| ② | Multiply $ \dfrac{x+4}{36} $ by $ x+1 $ to get $ \dfrac{x^2+5x+4}{36} $. Step 1: Write $ x+1 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{x+4}{36} \cdot x+1 & \xlongequal{\text{Step 1}} \frac{x+4}{36} \cdot \frac{x+1}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ \left( x+4 \right) \cdot \left( x+1 \right) }{ 36 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ x^2+x+4x+4 }{ 36 } = \frac{x^2+5x+4}{36} \end{aligned} $$ |
| ③ | Multiply $ \dfrac{x^2+5x+4}{36} $ by $ x-1 $ to get $ \dfrac{x^3+4x^2-x-4}{36} $. Step 1: Write $ x-1 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{x^2+5x+4}{36} \cdot x-1 & \xlongequal{\text{Step 1}} \frac{x^2+5x+4}{36} \cdot \frac{x-1}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ \left( x^2+5x+4 \right) \cdot \left( x-1 \right) }{ 36 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ x^3-x^2+5x^2-5x+4x-4 }{ 36 } = \frac{x^3+4x^2-x-4}{36} \end{aligned} $$ |
| ④ | Multiply $ \dfrac{x^3+4x^2-x-4}{36} $ by $ x^2-6x+9 $ to get $ \dfrac{x^5-2x^4-16x^3+38x^2+15x-36}{36} $. Step 1: Write $ x^2-6x+9 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{x^3+4x^2-x-4}{36} \cdot x^2-6x+9 & \xlongequal{\text{Step 1}} \frac{x^3+4x^2-x-4}{36} \cdot \frac{x^2-6x+9}{\color{red}{1}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( x^3+4x^2-x-4 \right) \cdot \left( x^2-6x+9 \right) }{ 36 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ x^5-6x^4+9x^3+4x^4-24x^3+36x^2-x^3+6x^2-9x-4x^2+24x-36 }{ 36 } = \\[1ex] &= \frac{x^5-2x^4-16x^3+38x^2+15x-36}{36} \end{aligned} $$ |