Find $ \left(1+x\right)^2 $ using formula.

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

where $ A = \color{blue}{ 1 } $ and $ B = \color{red}{ x }$.

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

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

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

Multiply $ \color{blue}{4x^2} $ by $ \left( 1+2x+x^2\right) $ $$ \color{blue}{4x^2} \cdot \left( 1+2x+x^2\right) = 4x^2+8x^3+4x^4 $$

Remove the parentheses by changing the sign of each term within them.

$$ - \left( 4x^2+4x+1 \right) = -4x^2-4x-1 $$

Combine like terms:

$$ \, \color{blue}{ \cancel{4x^2}} \,+8x^3+4x^4 \, \color{blue}{ -\cancel{4x^2}} \,-4x-1 = 4x^4+8x^3-4x-1 $$