We are tasked with finding $g(f(h(-3)))$, where the functions $f(x)$, $g(x)$, and $h(x)$ are defined as follows:

• $f(x) = 2x^2 + 3x - 5$
• $g(x) = 3x^2 - 10$
• $h(x) = 2x + 4$

Let's solve step by step.

Step 1: Find $h(-3)$

We start with $h(x) = 2x + 4$.

Substitute $x = -3$:

$h(-3) = 2(-3) + 4 = -6 + 4 = -2$

Step 2: Find $f(h(-3))$

Now, substitute $h(-3) = -2$ into $f(x) = 2x^2 + 3x - 5$:

$f(-2) = 2(-2)^2 + 3(-2) - 5 = 2(4) - 6 - 5 = 8 - 6 - 5 = -3$

Step 3: Find $g(f(h(-3)))$

Substitute $f(-2) = -3$ into $g(x) = 3x^2 - 10$:

$g(-3) = 3(-3)^2 - 10 = 3(9) - 10 = 27 - 10 = 17$

Final Answer:

$g(f(h(-3))) = 17$

Would you like a detailed breakdown of any part of the process, or have any questions?

Here are 5 related questions to extend your understanding:

1. What is the composition $f(g(x))$ with the same functions?
2. How do you compute $h(f(g(x)))$?
3. What happens when you evaluate $g(h(-3))$?
4. Can you find $g(f(h(0)))$?
5. How would you interpret these compositions graphically?

Tip: When solving composite functions, always work from the innermost function outwards.