https://file.aiquickdraw.com/mathbot/file/1732306265778y29ixn0o.jpg

To solve the problem in the image, we need to calculate \( (h \circ g)(n) \), which means \( h(g(n)) \). Let me break it down step by step:

### Given:
1. \( h(n) = -3n^2 + 3 \)
2. \( g(n) = 3n + 4 \)

### Step 1: Substitute \( g(n) \) into \( h(n) \)
To find \( h(g(n)) \), substitute \( g(n) = 3n + 4 \) into \( h(n) \):
\[
h(g(n)) = -3(g(n))^2 + 3
\]

### Step 2: Expand \( g(n) \) inside \( h(n) \)
Substitute \( g(n) = 3n + 4 \):
\[
h(g(n)) = -3(3n + 4)^2 + 3
\]

### Step 3: Expand \( (3n + 4)^2 \)
Using the formula \( (a + b)^2 = a^2 + 2ab + b^2 \):
\[
(3n + 4)^2 = (3n)^2 + 2(3n)(4) + 4^2
\]
\[
(3n + 4)^2 = 9n^2 + 24n + 16
\]

### Step 4: Substitute the expansion into \( h(g(n)) \)
\[
h(g(n)) = -3(9n^2 + 24n + 16) + 3
\]

### Step 5: Distribute the \(-3\) across the terms
\[
h(g(n)) = -27n^2 - 72n - 48 + 3
\]

### Step 6: Combine constants
\[
h(g(n)) = -27n^2 - 72n - 45
\]

### Final Answer:
\[
(h \circ g)(n) = -27n^2 - 72n - 45
\]

If you have questions or want more explanation, feel free to ask!

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### Related Questions:
1. What is \( (g \circ h)(n) \) based on the given functions?
2. How do you compute the domain of \( (h \circ g)(n) \)?
3. Can we simplify \( (h \circ g)(n) \) further using factoring?
4. What is the significance of composition in mathematical functions?
5. How can the method used here apply to functions with three compositions like \( (f \circ g \circ h)(n) \)?

### Tip:
When solving compositions of functions, always carefully substitute and expand terms step by step to avoid algebraic errors!

Given the functions h(n) = -3n^2 + 3 and g(n) = 3n + 4, find (h ∘ g)(n).

Learn how to solve the function composition (h ∘ g)(n) given h(n) = -3n^2 + 3 and g(n) = 3n + 4. This step-by-step solution demonstrates polynomial expansion and substitution, making it perfect for high school algebra students.

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