We are given the function $h(x) = -3x + 4$ with the domain $-2 \leq x < 5$, and we need to find its range.

Step 1: Find the values of $h(x)$ at the endpoints of the domain.

1. At $x = -2$: $h(-2) = -3(-2) + 4 = 6 + 4 = 10$ So, $h(x) = 10$ when $x = -2$.

2. At $x = 5$ (not included, but needed to find the boundary): $h(5) = -3(5) + 4 = -15 + 4 = -11$ So, $h(x) = -11$ as $x$ approaches 5 from the left.

Step 2: Determine the range.

The function $h(x) = -3x + 4$ is a linear function with a negative slope, meaning it decreases as $x$ increases. Therefore, as $x$ ranges from $-2$ to just below 5, $h(x)$ ranges from $10$ to just above $-11$.

Hence, the range of $h(x)$ is: $-11 < h(x) \leq 10$

Thus, the correct answer is the second option: $-11 < h(x) \leq 10$.

Would you like more details on any part of the solution?

Here are 5 related questions to expand on this concept:

1. How do you find the range of a quadratic function on a given domain?
2. What is the general method for finding the range of a function over an interval?
3. How does the slope of a linear function affect its range?
4. Can a linear function have a range that includes infinity?
5. How would the range change if the domain of $h(x)$ were extended to $-3 \leq x \leq 6$?

Tip: Always check the behavior of the function at the endpoints of the domain to determine its range correctly.