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We are given two functions, \( f(x) = 3x^3 - 5x + 3 \) defined over the interval \( -3 \leq x < 5 \), and \( g(x) = 2x^3 - 10x \) defined over \( -4 < x \leq 2 \).

We are tasked with finding \( (g - f)(x) \), which is computed as:

\[
(g - f)(x) = g(x) - f(x)
\]

Let’s calculate \( g(x) - f(x) \):

### Step 1: Subtract the functions
\[
(g - f)(x) = (2x^3 - 10x) - (3x^3 - 5x + 3)
\]

Distribute the negative sign:

\[
(g - f)(x) = 2x^3 - 10x - 3x^3 + 5x - 3
\]

### Step 2: Combine like terms
\[
(g - f)(x) = (2x^3 - 3x^3) + (-10x + 5x) - 3
\]

Simplifying the terms:

\[
(g - f)(x) = -x^3 - 5x - 3
\]

### Conclusion:
Thus, \( (g - f)(x) = -x^3 - 5x - 3 \), which matches the first option in the given choices.

### Answer:
\[
\boxed{(g - f)(x) = -x^3 - 5x - 3}
\]

Do you need further clarification or details on any steps? Here are five related questions:

1. How would the subtraction change if the signs in \( g(x) \) were different?
2. What is the domain of the function \( (g - f)(x) \)?
3. How would you graph \( (g - f)(x) \)?
4. What happens to \( (g - f)(x) \) at the boundaries of the intervals for \( f(x) \) and \( g(x) \)?
5. How would you calculate \( (g + f)(x) \) instead of \( (g - f)(x) \)?

**Tip:** Always be mindful of distributing negative signs when subtracting functions.

Sean f(x) = 3x^3 - 5x + 3, −3 ≤ x < 5, g(x) = 2x^3 - 10x, −4 < x ≤ 2. Hallar (g − f)(x).

Learn how to subtract polynomial functions by solving g(x) - f(x) where f(x) = 3x^3 - 5x + 3 and g(x) = 2x^3 - 10x. This detailed step-by-step solution covers function subtraction, distribution of negative signs, and combining like terms.

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