Find $ \left(x+2\right)^2 $ using formula.

$$ (A + B)^2 = \color{blue}{A^2} + 2 \cdot A \cdot B + \color{red}{B^2} $$

where $ A = \color{blue}{ x } $ and $ B = \color{red}{ 2 }$.

$$ \begin{aligned}\left(x+2\right)^2 = \color{blue}{x^2} +2 \cdot x \cdot 2 + \color{red}{2^2} = x^2+4x+4\end{aligned} $$

Find $ \left(x-4\right)^3 $ using formula

$$ (A - B) = A^3 - 3A^2B + 3AB^2 - B^3 $$

where $ A = x $ and $ B = 4 $.

$$ \left(x-4\right)^3 = x^3-3 \cdot x^2 \cdot 4 + 3 \cdot x \cdot 4^2-4^3 = x^3-12x^2+48x-64 $$

Multiply $ \dfrac{1}{4} $ by $ x $ to get $ \dfrac{ x }{ 4 } $.

Step 1: Write $ x $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{1}{4} \cdot x & \xlongequal{\text{Step 1}} \frac{1}{4} \cdot \frac{x}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 1 \cdot x }{ 4 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ x }{ 4 } \end{aligned} $$

Multiply $ \dfrac{x}{4} $ by $ x^2+4x+4 $ to get $ \dfrac{ x^3+4x^2+4x }{ 4 } $.

Step 1: Write $ x^2+4x+4 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{x}{4} \cdot x^2+4x+4 & \xlongequal{\text{Step 1}} \frac{x}{4} \cdot \frac{x^2+4x+4}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ x \cdot \left( x^2+4x+4 \right) }{ 4 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ x^3+4x^2+4x }{ 4 } \end{aligned} $$

Multiply $ \dfrac{x^3+4x^2+4x}{4} $ by $ x^3-12x^2+48x-64 $ to get $ \dfrac{x^6-8x^5+4x^4+80x^3-64x^2-256x}{4} $.

Step 1: Write $ x^3-12x^2+48x-64 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{x^3+4x^2+4x}{4} \cdot x^3-12x^2+48x-64 & \xlongequal{\text{Step 1}} \frac{x^3+4x^2+4x}{4} \cdot \frac{x^3-12x^2+48x-64}{\color{red}{1}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( x^3+4x^2+4x \right) \cdot \left( x^3-12x^2+48x-64 \right) }{ 4 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ x^6-12x^5+ \cancel{48x^4}-64x^3+4x^5 -\cancel{48x^4}+192x^3-256x^2+4x^4-48x^3+192x^2-256x }{ 4 } = \\[1ex] &= \frac{x^6-8x^5+4x^4+80x^3-64x^2-256x}{4} \end{aligned} $$

Multiply $ \dfrac{x^6-8x^5+4x^4+80x^3-64x^2-256x}{4} $ by $ x+5 $ to get $ \dfrac{x^7-3x^6-36x^5+100x^4+336x^3-576x^2-1280x}{4} $.

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.

$$ \begin{aligned} \frac{x^6-8x^5+4x^4+80x^3-64x^2-256x}{4} \cdot x+5 & \xlongequal{\text{Step 1}} \frac{x^6-8x^5+4x^4+80x^3-64x^2-256x}{4} \cdot \frac{x+5}{\color{red}{1}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( x^6-8x^5+4x^4+80x^3-64x^2-256x \right) \cdot \left( x+5 \right) }{ 4 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ x^7+5x^6-8x^6-40x^5+4x^5+20x^4+80x^4+400x^3-64x^3-320x^2-256x^2-1280x }{ 4 } = \\[1ex] &= \frac{x^7-3x^6-36x^5+100x^4+336x^3-576x^2-1280x}{4} \end{aligned} $$

Multiply $ \dfrac{x^7-3x^6-36x^5+100x^4+336x^3-576x^2-1280x}{4} $ by $ x-1 $ to get $ \dfrac{x^8-4x^7-33x^6+136x^5+236x^4-912x^3-704x^2+1280x}{4} $.

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.

$$ \begin{aligned} \frac{x^7-3x^6-36x^5+100x^4+336x^3-576x^2-1280x}{4} \cdot x-1 & \xlongequal{\text{Step 1}} \frac{x^7-3x^6-36x^5+100x^4+336x^3-576x^2-1280x}{4} \cdot \frac{x-1}{\color{red}{1}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( x^7-3x^6-36x^5+100x^4+336x^3-576x^2-1280x \right) \cdot \left( x-1 \right) }{ 4 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ x^8-x^7-3x^7+3x^6-36x^6+36x^5+100x^5-100x^4+336x^4-336x^3-576x^3+576x^2-1280x^2+1280x }{ 4 } = \\[1ex] &= \frac{x^8-4x^7-33x^6+136x^5+236x^4-912x^3-704x^2+1280x}{4} \end{aligned} $$

Multiply $ \dfrac{x^8-4x^7-33x^6+136x^5+236x^4-912x^3-704x^2+1280x}{4} $ by $ x+3 $ to get $ \dfrac{x^9-x^8-45x^7+37x^6+644x^5-204x^4-3440x^3-832x^2+3840x}{4} $.

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.

$$ \begin{aligned} \frac{x^8-4x^7-33x^6+136x^5+236x^4-912x^3-704x^2+1280x}{4} \cdot x+3 & \xlongequal{\text{Step 1}} \frac{x^8-4x^7-33x^6+136x^5+236x^4-912x^3-704x^2+1280x}{4} \cdot \frac{x+3}{\color{red}{1}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( x^8-4x^7-33x^6+136x^5+236x^4-912x^3-704x^2+1280x \right) \cdot \left( x+3 \right) }{ 4 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ x^9+3x^8-4x^8-12x^7-33x^7-99x^6+136x^6+408x^5+236x^5+708x^4-912x^4-2736x^3-704x^3-2112x^2+1280x^2+3840x }{ 4 } = \\[1ex] &= \frac{x^9-x^8-45x^7+37x^6+644x^5-204x^4-3440x^3-832x^2+3840x}{4} \end{aligned} $$