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

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 }$.

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

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

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

Combine like terms:

$$ 1+ \color{blue}{2x} + \color{red}{x^2} + \color{blue}{2x} + \color{green}{4x^2} + \color{orange}{2x^3} + \color{green}{x^2} + \color{orange}{2x^3} +x^4 = x^4+ \color{orange}{4x^3} + \color{green}{6x^2} + \color{blue}{4x} +1 $$

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 }$.

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

Multiply $ \color{blue}{4} $ by $ \left( x^4+4x^3+6x^2+4x+1\right) $ $$ \color{blue}{4} \cdot \left( x^4+4x^3+6x^2+4x+1\right) = 4x^4+16x^3+24x^2+16x+4 $$Multiply $ \color{blue}{17x^2} $ by $ \left( 1+2x+x^2\right) $ $$ \color{blue}{17x^2} \cdot \left( 1+2x+x^2\right) = 17x^2+34x^3+17x^4 $$Multiply $ \color{blue}{132x} $ by $ \left( x+1\right) $ $$ \color{blue}{132x} \cdot \left( x+1\right) = 132x^2+132x $$

Combine like terms:

$$ \color{blue}{4x^4} + \color{blue}{4x^4} +16x^3+24x^2+16x+4 = \color{blue}{8x^4} +16x^3+24x^2+16x+4 $$

Combine like terms:

$$ \color{blue}{8x^4} + \color{red}{16x^3} + \color{green}{24x^2} +16x+4+ \color{green}{17x^2} + \color{red}{34x^3} + \color{blue}{17x^4} = \\ = \color{blue}{25x^4} + \color{red}{50x^3} + \color{green}{41x^2} +16x+4 $$

Multiply each term of $ \left( \color{blue}{132x^2+132x}\right) $ by each term in $ \left( 2x^2+2x+1\right) $.

$$ \left( \color{blue}{132x^2+132x}\right) \cdot \left( 2x^2+2x+1\right) = 264x^4+264x^3+132x^2+264x^3+264x^2+132x $$

Combine like terms:

$$ 264x^4+ \color{blue}{264x^3} + \color{red}{132x^2} + \color{blue}{264x^3} + \color{red}{264x^2} +132x = 264x^4+ \color{blue}{528x^3} + \color{red}{396x^2} +132x $$

Combine like terms:

$$ \color{blue}{25x^4} + \color{red}{50x^3} + \color{green}{41x^2} + \color{orange}{16x} +4+ \color{blue}{264x^4} + \color{red}{528x^3} + \color{green}{396x^2} + \color{orange}{132x} = \\ = \color{blue}{289x^4} + \color{red}{578x^3} + \color{green}{437x^2} + \color{orange}{148x} +4 $$