Multiply $ \color{blue}{12x^4} $ by $ \left( x-3\right) $ $$ \color{blue}{12x^4} \cdot \left( x-3\right) = 12x^5-36x^4 $$

Multiply $ \color{blue}{30} $ by $ \left( x+3\right) $ $$ \color{blue}{30} \cdot \left( x+3\right) = 30x+90 $$

Multiply each term of $ \left( \color{blue}{30x+90}\right) $ by each term in $ \left( x+5\right) $.

$$ \left( \color{blue}{30x+90}\right) \cdot \left( x+5\right) = 30x^2+150x+90x+450 $$

Combine like terms:

$$ 30x^2+ \color{blue}{150x} + \color{blue}{90x} +450 = 30x^2+ \color{blue}{240x} +450 $$

Simplify $ \dfrac{x+5}{30x^2+240x+450} $ to $ \dfrac{1}{30x+90} $.

Factor both the denominator and the numerator, then cancel the common factor. $\color{blue}{x+5}$.

$$ \begin{aligned} \frac{x+5}{30x^2+240x+450} & =\frac{ 1 \cdot \color{blue}{ \left( x+5 \right) }}{ \left( 30x+90 \right) \cdot \color{blue}{ \left( x+5 \right) }} = \\[1ex] &= \frac{1}{30x+90} \end{aligned} $$

Multiply $12x^5-36x^4$ by $ \dfrac{1}{30x+90} $ to get $ \dfrac{ 2x^5-6x^4 }{ 5x+15 } $.

Step 1: Write $ 12x^5-36x^4 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Factor numerators and denominators.

Step 3: Cancel common factors.

Step 4: Multiply numerators and denominators.

Step 5: Simplify numerator and denominator.

$$ \begin{aligned} 12x^5-36x^4 \cdot \frac{1}{30x+90} & \xlongequal{\text{Step 1}} \frac{12x^5-36x^4}{\color{red}{1}} \cdot \frac{1}{30x+90} \xlongequal{\text{Step 2}} \frac{ \left( 2x^5-6x^4 \right) \cdot \color{blue}{6} }{ 1 } \cdot \frac{ 1 }{ \left( 5x+15 \right) \cdot \color{blue}{6} } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 2x^5-6x^4 }{ 1 } \cdot \frac{ 1 }{ 5x+15 } \xlongequal{\text{Step 4}} \frac{ \left( 2x^5-6x^4 \right) \cdot 1 }{ 1 \cdot \left( 5x+15 \right) } = \\[1ex] & \xlongequal{\text{Step 5}} \frac{ 2x^5-6x^4 }{ 5x+15 } \end{aligned} $$