Multiply $10u$ by $ \dfrac{z^4}{2} $ to get $ \dfrac{ 10uz^4 }{ 2 } $.

Step 1: Write $ 10u $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 10u \cdot \frac{z^4}{2} & \xlongequal{\text{Step 1}} \frac{10u}{\color{red}{1}} \cdot \frac{z^4}{2} \xlongequal{\text{Step 2}} \frac{ 10u \cdot z^4 }{ 1 \cdot 2 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 10uz^4 }{ 2 } \end{aligned} $$

Multiply $ \dfrac{10uz^4}{2} $ by $ u^2 $ to get $ \dfrac{ 10u^3z^4 }{ 2 } $.

Step 1: Write $ u^2 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{10uz^4}{2} \cdot u^2 & \xlongequal{\text{Step 1}} \frac{10uz^4}{2} \cdot \frac{u^2}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 10uz^4 \cdot u^2 }{ 2 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 10u^3z^4 }{ 2 } \end{aligned} $$

Multiply $ \dfrac{10u^3z^4}{2} $ by $ z^5 $ to get $ \dfrac{ 10u^3z^9 }{ 2 } $.

Step 1: Write $ z^5 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{10u^3z^4}{2} \cdot z^5 & \xlongequal{\text{Step 1}} \frac{10u^3z^4}{2} \cdot \frac{z^5}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 10u^3z^4 \cdot z^5 }{ 2 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 10u^3z^9 }{ 2 } \end{aligned} $$

Subtract $ \dfrac{10u^3z^9}{2} $ from $ -4u^3z^7+12u^7z^7 $ to get $ \dfrac{ \color{purple}{ 24u^7z^7-10u^3z^9-8u^3z^7 } }{ 2 }$.

Step 1: Write $ -4u^3z^7+12u^7z^7 $ 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 first fraction by $\color{blue}{ 2 }$.

$$ \begin{aligned} -4u^3z^7+12u^7z^7- \frac{10u^3z^9}{2} & \xlongequal{\text{Step 1}} \frac{-4u^3z^7+12u^7z^7}{\color{red}{1}} - \frac{10u^3z^9}{2} = \\[1ex] &= \frac{ \left( -4u^3z^7+12u^7z^7 \right) \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} - \frac{ 10u^3z^9 }{ 2 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ -8u^3z^7+24u^7z^7 } }{ 2 } - \frac{ \color{purple}{ 10u^3z^9 } }{ 2 }=\frac{ \color{purple}{ 24u^7z^7-10u^3z^9-8u^3z^7 } }{ 2 } \end{aligned} $$