To find the area of a rectangle, we multiply its length and width. Here, the expressions for the sides of the rectangle are \( 3x^3 \) and \( -4x^3 + 5 \).
Let's calculate the area:
\[ \text{Area} = \text{Length} \times \text{Width} = (3x^3)(-4x^3 + 5) \]
Now, distribute \( 3x^3 \) across the two terms in the second expression:
\[ = 3x^3 \cdot (-4x^3) + 3x^3 \cdot 5 \]
Calculating each term separately:
- \( 3x^3 \cdot (-4x^3) = -12x^{6} \)
- \( 3x^3 \cdot 5 = 15x^3 \)
Now, combine these results:
\[ \text{Area} = -12x^{6} + 15x^3 \]
So, the area of the rectangle is
\[ -12x^{6} + 15x^3 \]
The answer is option C) \(-12x^6 + 15x^3\).