To find the interval where the function \(f(x) = 10 - 6x - 2x^2\) is decreasing, we need to determine where its derivative \(f'(x)\) is less than zero. First, find the derivative:
1. Differentiate the function: \(f(x) = 10 - 6x - 2x^2\).
\[f'(x) = \frac{d}{dx}(10) - \frac{d}{dx}(6x) - \frac{d}{dx}(2x^2)\]
2. Calculate each term: \[\frac{d}{dx}(10) = 0\], \[\frac{d}{dx}(6x) = 6\], \[\frac{d}{dx}(2x^2) = 4x\]
3. Thus, \(f'(x) = 0 - 6 - 4x = -6 - 4x\).
We set \(f'(x) < 0\) to find where the function is decreasing:
\[-6 - 4x < 0\]
4. Solve this inequality:
\[-4x < 6\]
\[x > \frac{-6}{4}\]
\[x > \frac{-3}{2}\]
Therefore, the function is decreasing in the interval \((\frac{-3}{2}, \infty)\).
This corresponds to the correct option: \((\frac{-3}{2}, \infty)\).