Subtract $ \dfrac{x^4}{4} $ from $ -3x^2-x^3 $ to get $ \dfrac{ \color{purple}{ -x^4-4x^3-12x^2 } }{ 4 }$.

Step 1: Write $ -3x^2-x^3 $ 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 $\color{blue}{ 4 }$.

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