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

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

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

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

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

Combine like terms:

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

Find $ \left(1+x\right)^3 $ using formula

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

where $ A = 1 $ and $ B = x $.

$$ \left(1+x\right)^3 = 1^3+3 \cdot 1^2 \cdot x + 3 \cdot 1 \cdot x^2+x^3 = 1+3x+3x^2+x^3 $$

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}{132} $ by $ \left( 1+3x+3x^2+x^3\right) $ $$ \color{blue}{132} \cdot \left( 1+3x+3x^2+x^3\right) = 132+396x+396x^2+132x^3 $$Multiply $ \color{blue}{17} $ by $ \left( 1+2x+x^2\right) $ $$ \color{blue}{17} \cdot \left( 1+2x+x^2\right) = 17+34x+17x^2 $$Multiply $ \color{blue}{132} $ by $ \left( 1+x\right) $ $$ \color{blue}{132} \cdot \left( 1+x\right) = 132+132x $$

Combine like terms:

$$ 4x^4+ \color{blue}{16x^3} + \color{red}{24x^2} + \color{green}{16x} + \color{orange}{4} + \color{orange}{132} + \color{green}{396x} + \color{red}{396x^2} + \color{blue}{132x^3} = \\ = 4x^4+ \color{blue}{148x^3} + \color{red}{420x^2} + \color{green}{412x} + \color{orange}{136} $$

Combine like terms:

$$ 4x^4+148x^3+ \color{blue}{420x^2} + \color{red}{412x} + \color{green}{136} + \color{green}{17} + \color{red}{34x} + \color{blue}{17x^2} = \\ = 4x^4+148x^3+ \color{blue}{437x^2} + \color{red}{446x} + \color{green}{153} $$

Combine like terms:

$$ 4x^4+148x^3+437x^2+ \color{blue}{446x} + \color{red}{153} + \color{red}{132} + \color{blue}{132x} = 4x^4+148x^3+437x^2+ \color{blue}{578x} + \color{red}{285} $$

Combine like terms:

$$ 4x^4+148x^3+437x^2+578x+ \color{blue}{285} + \color{blue}{4} = 4x^4+148x^3+437x^2+578x+ \color{blue}{289} $$