Multiply each term of $ \left( \color{blue}{2z-ip}\right) $ by each term in $ \left( 2z+2\right) $.

$$ \left( \color{blue}{2z-ip}\right) \cdot \left( 2z+2\right) = 4z^2+4z-2ipz-2ip $$

Multiply $ \color{blue}{2glo} $ by $ \left( z^2-2z+3\right) $ $$ \color{blue}{2glo} \cdot \left( z^2-2z+3\right) = 2gloz^2-4gloz+6glo $$

Subtract $2gloz^2-4gloz+6glo$ from $ \dfrac{4z^2+4z-2ipz-2ip}{z^2+2z+3} $ to get $ \dfrac{ \color{purple}{ -2gloz^4-4gloz^2-18glo-2ipz-2ip+4z^2+4z } }{ z^2+2z+3 }$.

Step 1: Write $ 2gloz^2-4gloz+6glo $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{z^2+2z+3}$.

$$ \begin{aligned} \frac{4z^2+4z-2ipz-2ip}{z^2+2z+3} -2gloz^2-4gloz+6glo & \xlongequal{\text{Step 1}} \frac{4z^2+4z-2ipz-2ip}{z^2+2z+3} - \frac{2gloz^2-4gloz+6glo}{\color{red}{1}} = \\[1ex] &= \frac{ 4z^2+4z-2ipz-2ip }{ z^2+2z+3 } - \frac{ \left( 2gloz^2-4gloz+6glo \right) \cdot \color{blue}{ \left( z^2+2z+3 \right) }}{ 1 \cdot \color{blue}{ \left( z^2+2z+3 \right) }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 4z^2+4z-2ipz-2ip } }{ z^2+2z+3 } - \frac{ \color{purple}{ 2gloz^4+ \cancel{4gloz^3}+6gloz^2 -\cancel{4gloz^3}-8gloz^2 -\cancel{12gloz}+6gloz^2+ \cancel{12gloz}+18glo } }{ z^2+2z+3 } = \\[1ex] &=\frac{ \color{purple}{ 4z^2+4z-2ipz-2ip - \left( 2gloz^4+ \cancel{4gloz^3}+6gloz^2 -\cancel{4gloz^3}-8gloz^2 -\cancel{12gloz}+6gloz^2+ \cancel{12gloz}+18glo \right) } }{ z^2+2z+3 } = \\[1ex] &=\frac{ \color{purple}{ -2gloz^4-4gloz^2-18glo-2ipz-2ip+4z^2+4z } }{ z^2+2z+3 } \end{aligned} $$

Combine like terms:

$$ 4z^2 \color{blue}{-4ipz} + \color{blue}{ipz} = \color{blue}{-3ipz} +4z^2 $$

Divide $ \dfrac{-2gloz^4-4gloz^2-18glo-2ipz-2ip+4z^2+4z}{z^2+2z+3} $ by $ -3ipz+4z^2 $ to get $ \dfrac{-2gloz^4-4gloz^2-18glo-2ipz-2ip+4z^2+4z}{-3ipz^3-6ipz^2+4z^4-9ipz+8z^3+12z^2} $.

Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{ \frac{-2gloz^4-4gloz^2-18glo-2ipz-2ip+4z^2+4z}{z^2+2z+3} }{-3ipz+4z^2} & \xlongequal{\text{Step 1}} \frac{-2gloz^4-4gloz^2-18glo-2ipz-2ip+4z^2+4z}{z^2+2z+3} \cdot \frac{\color{blue}{1}}{\color{blue}{-3ipz+4z^2}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( -2gloz^4-4gloz^2-18glo-2ipz-2ip+4z^2+4z \right) \cdot 1 }{ \left( z^2+2z+3 \right) \cdot \left( -3ipz+4z^2 \right) } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ -2gloz^4-4gloz^2-18glo-2ipz-2ip+4z^2+4z }{ -3ipz^3+4z^4-6ipz^2+8z^3-9ipz+12z^2 } = \frac{-2gloz^4-4gloz^2-18glo-2ipz-2ip+4z^2+4z}{-3ipz^3-6ipz^2+4z^4-9ipz+8z^3+12z^2} \end{aligned} $$