Find $ \left(x-2\right)^3 $ using formula

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

where $ A = x $ and $ B = 2 $.

$$ \left(x-2\right)^3 = x^3-3 \cdot x^2 \cdot 2 + 3 \cdot x \cdot 2^2-2^3 = x^3-6x^2+12x-8 $$

Multiply each term of $ \left( \color{blue}{x^3-6x^2+12x-8}\right) $ by each term in $ \left( x+2i\right) $.

$$ \left( \color{blue}{x^3-6x^2+12x-8}\right) \cdot \left( x+2i\right) = x^4+2ix^3-6x^3-12ix^2+12x^2+24ix-8x-16i $$

Combine like terms:

$$ x^4+2ix^3-6x^3-12ix^2+12x^2+24ix-8x-16i = 2ix^3+x^4-12ix^2-6x^3+24ix+12x^2-16i-8x $$

Multiply each term of $ \left( \color{blue}{2ix^3+x^4-12ix^2-6x^3+24ix+12x^2-16i-8x}\right) $ by each term in $ \left( x-2i\right) $.

$$ \left( \color{blue}{2ix^3+x^4-12ix^2-6x^3+24ix+12x^2-16i-8x}\right) \cdot \left( x-2i\right) = \\ = \cancel{2ix^4}-4i^2x^3+x^5 -\cancel{2ix^4} -\cancel{12ix^3}+24i^2x^2-6x^4+ \cancel{12ix^3}+ \cancel{24ix^2}-48i^2x+12x^3 -\cancel{24ix^2} -\cancel{16ix}+32i^2-8x^2+ \cancel{16ix} $$

Combine like terms:

$$ \, \color{blue}{ \cancel{2ix^4}} \,-4i^2x^3+x^5 \, \color{blue}{ -\cancel{2ix^4}} \, \, \color{green}{ -\cancel{12ix^3}} \,+24i^2x^2-6x^4+ \, \color{green}{ \cancel{12ix^3}} \,+ \, \color{blue}{ \cancel{24ix^2}} \,-48i^2x+12x^3 \, \color{blue}{ -\cancel{24ix^2}} \, \, \color{green}{ -\cancel{16ix}} \,+32i^2-8x^2+ \, \color{green}{ \cancel{16ix}} \, = -4i^2x^3+x^5+24i^2x^2-6x^4-48i^2x+12x^3+32i^2-8x^2 $$