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Let's analyze the given problem step by step.

We are given the function:
\[
f(x) = 8 + x + x^2
\]
and \( h \neq 0 \).

### (a) Does \( f(7 + 1) = f(7) + f(1) \)?
We need to evaluate both sides.

- \( f(7 + 1) = f(8) \)
  - \( f(8) = 8 + 8 + 8^2 = 8 + 8 + 64 = 80 \)

- Now, calculate \( f(7) + f(1) \):
  - \( f(7) = 8 + 7 + 7^2 = 8 + 7 + 49 = 64 \)
  - \( f(1) = 8 + 1 + 1^2 = 8 + 1 + 1 = 10 \)
  - So, \( f(7) + f(1) = 64 + 10 = 74 \)

Clearly, \( f(8) \neq f(7) + f(1) \), so the answer is **No**.

---

### (b) Find \( f(x + h) \).

We need to substitute \( x + h \) into the function \( f(x) = 8 + x + x^2 \):
\[
f(x + h) = 8 + (x + h) + (x + h)^2
\]
Expanding the expression:
\[
f(x + h) = 8 + x + h + (x^2 + 2xh + h^2)
\]
Simplifying:
\[
f(x + h) = 8 + x + h + x^2 + 2xh + h^2
\]
Thus, \( f(x + h) = 8 + x + x^2 + h + 2xh + h^2 \).

---

### (c) Does \( f(x + h) = f(x) + f(h) \)?
Let's calculate both sides.

- \( f(x + h) = 8 + x + x^2 + h + 2xh + h^2 \) (from part b).
  
- Now, calculate \( f(x) + f(h) \):
  - \( f(x) = 8 + x + x^2 \)
  - \( f(h) = 8 + h + h^2 \)
  - So, \( f(x) + f(h) = (8 + x + x^2) + (8 + h + h^2) = 16 + x + h + x^2 + h^2 \)

Clearly, \( f(x + h) \neq f(x) + f(h) \) because the term \( 2xh \) is missing on the right-hand side. So, the answer is **No**.

---

### (d) Does \( f(x + h) = f(x) + h \)?

Let's compare both sides.

- \( f(x + h) = 8 + x + x^2 + h + 2xh + h^2 \) (from part b).
- \( f(x) + h = (8 + x + x^2) + h = 8 + x + x^2 + h \)

Since \( f(x + h) \) contains additional terms \( 2xh \) and \( h^2 \), the answer is **No**.

---

### (e) Find \( \frac{f(x + h) - f(x)}{h} \) and simplify.

First, let's compute the difference:
\[
f(x + h) - f(x) = (8 + x + x^2 + h + 2xh + h^2) - (8 + x + x^2)
\]
Simplifying:
\[
f(x + h) - f(x) = h + 2xh + h^2
\]
Now, divide by \( h \):
\[
\frac{f(x + h) - f(x)}{h} = \frac{h + 2xh + h^2}{h}
\]
Simplifying further:
\[
\frac{f(x + h) - f(x)}{h} = 1 + 2x + h
\]

Thus, the simplified form is \( 1 + 2x + h \).

---

Do you need further clarifications or details on any step? Here are 5 related questions to explore:

1. How does the behavior of \( f(x) \) change as \( h \to 0 \)?
2. What is the derivative of \( f(x) = 8 + x + x^2 \)?
3. Can \( f(x) \) be factored in any useful way for solving future problems?
4. How does the quadratic term \( x^2 \) impact the difference quotient in part (e)?
5. How would you solve the limit of the difference quotient as \( h \to 0 \)?

**Tip:** The difference quotient \( \frac{f(x+h) - f(x)}{h} \) is a crucial step in finding the derivative of a function!

Let f(x) = 8 + x + x^2 and h ≠ 0. Solve various properties of the function including finding values, differences, and analyzing equality relations.

Learn how to analyze the quadratic function f(x) = 8 + x + x^2. Solve questions involving function evaluation, the difference quotient, and polynomial simplification. This detailed solution is suitable for Grades 9-11 students.

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