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

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

Combine like terms:

$$ x^4+ \color{blue}{4x^2} + \color{blue}{x^2} +4 = x^4+ \color{blue}{5x^2} +4 $$

Multiply each term of $ \left( \color{blue}{x^4+5x^2+4}\right) $ by each term in $ \left( x^2+2x+2\right) $.

$$ \left( \color{blue}{x^4+5x^2+4}\right) \cdot \left( x^2+2x+2\right) = x^6+2x^5+2x^4+5x^4+10x^3+10x^2+4x^2+8x+8 $$

Combine like terms:

$$ x^6+2x^5+ \color{blue}{2x^4} + \color{blue}{5x^4} +10x^3+ \color{red}{10x^2} + \color{red}{4x^2} +8x+8 = x^6+2x^5+ \color{blue}{7x^4} +10x^3+ \color{red}{14x^2} +8x+8 $$

Multiply each term of $ \left( \color{blue}{x^6+2x^5+7x^4+10x^3+14x^2+8x+8}\right) $ by each term in $ \left( x^2-2x+2\right) $.

$$ \left( \color{blue}{x^6+2x^5+7x^4+10x^3+14x^2+8x+8}\right) \cdot \left( x^2-2x+2\right) = \\ = x^8 -\cancel{2x^7}+2x^6+ \cancel{2x^7}-4x^6+4x^5+7x^6-14x^5+14x^4+10x^5-20x^4+20x^3+14x^4-28x^3+28x^2+8x^3-16x^2+ \cancel{16x}+8x^2 -\cancel{16x}+16 $$

Combine like terms:

$$ x^8 \, \color{blue}{ -\cancel{2x^7}} \,+ \color{green}{2x^6} + \, \color{blue}{ \cancel{2x^7}} \, \color{orange}{-4x^6} + \color{blue}{4x^5} + \color{orange}{7x^6} \color{red}{-14x^5} + \color{green}{14x^4} + \color{red}{10x^5} \color{orange}{-20x^4} + \color{blue}{20x^3} + \color{orange}{14x^4} \color{red}{-28x^3} + \color{green}{28x^2} + \color{red}{8x^3} \color{orange}{-16x^2} + \, \color{blue}{ \cancel{16x}} \,+ \color{orange}{8x^2} \, \color{blue}{ -\cancel{16x}} \,+16 = \\ = x^8+ \color{orange}{5x^6} + \color{orange}{8x^4} + \color{orange}{20x^2} +16 $$