$$\dfrac{4}{45}(x-5)(x-3)(x-1)(x+1)(x+3)=\dfrac{4x^5-20x^4-40x^3+200x^2+36x-180}{45}$$
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{4}{45}(x-5)(x-3)(x-1)(x+1)(x+3)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{4x-20}{45}(x-3)(x-1)(x+1)(x+3) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{4x^2-32x+60}{45}(x-1)(x+1)(x+3) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{4x^3-36x^2+92x-60}{45}(x+1)(x+3) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{4x^4-32x^3+56x^2+32x-60}{45}(x+3) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{4x^5-20x^4-40x^3+200x^2+36x-180}{45}\end{aligned} $$ | |
| ① | Multiply $ \dfrac{4}{45} $ by $ x-5 $ to get $ \dfrac{ 4x-20 }{ 45 } $. Step 1: Write $ x-5 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{4}{45} \cdot x-5 & \xlongequal{\text{Step 1}} \frac{4}{45} \cdot \frac{x-5}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 4 \cdot \left( x-5 \right) }{ 45 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 4x-20 }{ 45 } \end{aligned} $$ |
| ② | Multiply $ \dfrac{4x-20}{45} $ by $ x-3 $ to get $ \dfrac{4x^2-32x+60}{45} $. Step 1: Write $ x-3 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{4x-20}{45} \cdot x-3 & \xlongequal{\text{Step 1}} \frac{4x-20}{45} \cdot \frac{x-3}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ \left( 4x-20 \right) \cdot \left( x-3 \right) }{ 45 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 4x^2-12x-20x+60 }{ 45 } = \frac{4x^2-32x+60}{45} \end{aligned} $$ |
| ③ | Multiply $ \dfrac{4x^2-32x+60}{45} $ by $ x-1 $ to get $ \dfrac{4x^3-36x^2+92x-60}{45} $. 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. $$ \begin{aligned} \frac{4x^2-32x+60}{45} \cdot x-1 & \xlongequal{\text{Step 1}} \frac{4x^2-32x+60}{45} \cdot \frac{x-1}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ \left( 4x^2-32x+60 \right) \cdot \left( x-1 \right) }{ 45 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 4x^3-4x^2-32x^2+32x+60x-60 }{ 45 } = \frac{4x^3-36x^2+92x-60}{45} \end{aligned} $$ |
| ④ | Multiply $ \dfrac{4x^3-36x^2+92x-60}{45} $ by $ x+1 $ to get $ \dfrac{4x^4-32x^3+56x^2+32x-60}{45} $. 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. $$ \begin{aligned} \frac{4x^3-36x^2+92x-60}{45} \cdot x+1 & \xlongequal{\text{Step 1}} \frac{4x^3-36x^2+92x-60}{45} \cdot \frac{x+1}{\color{red}{1}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( 4x^3-36x^2+92x-60 \right) \cdot \left( x+1 \right) }{ 45 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ 4x^4+4x^3-36x^3-36x^2+92x^2+92x-60x-60 }{ 45 } = \\[1ex] &= \frac{4x^4-32x^3+56x^2+32x-60}{45} \end{aligned} $$ |
| ⑤ | Multiply $ \dfrac{4x^4-32x^3+56x^2+32x-60}{45} $ by $ x+3 $ to get $ \dfrac{4x^5-20x^4-40x^3+200x^2+36x-180}{45} $. Step 1: Write $ x+3 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{4x^4-32x^3+56x^2+32x-60}{45} \cdot x+3 & \xlongequal{\text{Step 1}} \frac{4x^4-32x^3+56x^2+32x-60}{45} \cdot \frac{x+3}{\color{red}{1}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( 4x^4-32x^3+56x^2+32x-60 \right) \cdot \left( x+3 \right) }{ 45 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ 4x^5+12x^4-32x^4-96x^3+56x^3+168x^2+32x^2+96x-60x-180 }{ 45 } = \\[1ex] &= \frac{4x^5-20x^4-40x^3+200x^2+36x-180}{45} \end{aligned} $$ |