h(n) = n^2 + 4
g(n) = -2n -5

Find (3h + 2g)(n)


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To find \((3h + 2g)(n)\), we need to calculate the expression by combining the functions \(h(n)\) and \(g(n)\). Let's go through the steps carefully.

### Step 1: Start with the given functions
- \(h(n) = n^2 + 4\)
- \(g(n) = -2n - 5\)

### Step 2: Multiply \(h(n)\) by 3 and \(g(n)\) by 2
We need to first multiply each function by the given constants:

- \(3h(n) = 3(n^2 + 4) = 3n^2 + 12\)
- \(2g(n) = 2(-2n - 5) = -4n - 10\)

### Step 3: Add the results together
Now, sum the two expressions:

\[
3h(n) + 2g(n) = (3n^2 + 12) + (-4n - 10)
\]

Combine like terms:

\[
3h(n) + 2g(n) = 3n^2 - 4n + 2
\]

### Final Expression:
\[
(3h + 2g)(n) = 3n^2 - 4n + 2
\]

This is the simplified form of \((3h + 2g)(n)\).

Would you like further details or have any questions?

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Here are some related questions:

1. What is the value of \((3h + 2g)(n)\) when \(n = 1\)?
2. How would the expression change if the constants were different (e.g., \(4h + 3g\))?
3. Can you factor the quadratic expression \(3n^2 - 4n + 2\)?
4. How does this result relate to solving systems of equations?
5. What happens to the functions if you differentiate \((3h + 2g)(n)\)?

**Tip:** When combining functions, it's important to distribute constants carefully and combine like terms for simplification.

h(n) = n^2 + 4, g(n) = -2n - 5. Find (3h + 2g)(n).

Learn how to combine polynomial functions and find the expression (3h + 2g)(n) when h(n) = n^2 + 4 and g(n) = -2n - 5. This problem covers key algebra concepts including quadratic and linear functions, simplifying expressions, and polynomial combination. Suitable for students in Grades 9-12.

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