$$ (x^2+1)^4 = (x^2+1)^2 \cdot (x^2+1)^2 $$

Find $ \left(x^2+1\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^2 } $ and $ B = \color{red}{ 1 }$.

$$ \begin{aligned}\left(x^2+1\right)^2 = \color{blue}{\left( x^2 \right)^2} +2 \cdot x^2 \cdot 1 + \color{red}{1^2} = x^4+2x^2+1\end{aligned} $$

Multiply each term of $ \left( \color{blue}{x^4+2x^2+1}\right) $ by each term in $ \left( x^4+2x^2+1\right) $.

$$ \left( \color{blue}{x^4+2x^2+1}\right) \cdot \left( x^4+2x^2+1\right) = x^8+2x^6+x^4+2x^6+4x^4+2x^2+x^4+2x^2+1 $$

Combine like terms:

$$ x^8+ \color{blue}{2x^6} + \color{red}{x^4} + \color{blue}{2x^6} + \color{green}{4x^4} + \color{orange}{2x^2} + \color{green}{x^4} + \color{orange}{2x^2} +1 = \\ = x^8+ \color{blue}{4x^6} + \color{green}{6x^4} + \color{orange}{4x^2} +1 $$2x^2+2x^2=4x^2

Multiply $ \dfrac{x^2}{2} $ by $ x^8+4x^6+6x^4+4x^2+1 $ to get $ \dfrac{ x^{10}+4x^8+6x^6+4x^4+x^2 }{ 2 } $.

Step 1: Write $ x^8+4x^6+6x^4+4x^2+1 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{x^2}{2} \cdot x^8+4x^6+6x^4+4x^2+1 & \xlongequal{\text{Step 1}} \frac{x^2}{2} \cdot \frac{x^8+4x^6+6x^4+4x^2+1}{\color{red}{1}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ x^2 \cdot \left( x^8+4x^6+6x^4+4x^2+1 \right) }{ 2 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ x^{10}+4x^8+6x^6+4x^4+x^2 }{ 2 } \end{aligned} $$

Subtract $x^2+1$ from $ \dfrac{x^{10}+4x^8+6x^6+4x^4+x^2}{2} $ to get $ \dfrac{ \color{purple}{ x^{10}+4x^8+6x^6+4x^4-x^2-2 } }{ 2 }$.

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

Step 2: To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{2}$.

$$ \begin{aligned} \frac{x^{10}+4x^8+6x^6+4x^4+x^2}{2} -x^2+1 & \xlongequal{\text{Step 1}} \frac{x^{10}+4x^8+6x^6+4x^4+x^2}{2} - \frac{x^2+1}{\color{red}{1}} = \\[1ex] &= \frac{ x^{10}+4x^8+6x^6+4x^4+x^2 }{ 2 } - \frac{ \left( x^2+1 \right) \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x^{10}+4x^8+6x^6+4x^4+x^2 } }{ 2 } - \frac{ \color{purple}{ 2x^2+2 } }{ 2 }=\frac{ \color{purple}{ x^{10}+4x^8+6x^6+4x^4+x^2 - \left( 2x^2+2 \right) } }{ 2 } = \\[1ex] &=\frac{ \color{purple}{ x^{10}+4x^8+6x^6+4x^4-x^2-2 } }{ 2 } \end{aligned} $$