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We are given the function \( f(x) = x^2 + 6x + 5 \), and we need to evaluate it for different values of the independent variable. Let's go through each part:

### a. \( f(8) \)
We substitute \( x = 8 \) into the function \( f(x) \):
\[
f(8) = 8^2 + 6(8) + 5 = 64 + 48 + 5 = 117
\]
Thus, \( f(8) = 117 \).

### b. \( f(x + 5) \)
We substitute \( x + 5 \) in place of \( x \) in the function:
\[
f(x + 5) = (x + 5)^2 + 6(x + 5) + 5
\]
First, expand the terms:
\[
(x + 5)^2 = x^2 + 10x + 25
\]
\[
6(x + 5) = 6x + 30
\]
Now, sum everything:
\[
f(x + 5) = x^2 + 10x + 25 + 6x + 30 + 5 = x^2 + 16x + 60
\]
Thus, \( f(x + 5) = x^2 + 16x + 60 \).

### c. \( f(-x) \)
We substitute \( -x \) in place of \( x \):
\[
f(-x) = (-x)^2 + 6(-x) + 5
\]
Simplify each term:
\[
(-x)^2 = x^2
\]
\[
6(-x) = -6x
\]
Now, sum everything:
\[
f(-x) = x^2 - 6x + 5
\]
Thus, \( f(-x) = x^2 - 6x + 5 \).

### Summary:
- \( f(8) = 117 \)
- \( f(x + 5) = x^2 + 16x + 60 \)
- \( f(-x) = x^2 - 6x + 5 \)

Would you like more details or have any questions?

Here are some related questions to deepen your understanding:
1. What happens when you evaluate \( f(0) \)?
2. How would the function change if the coefficient of \( x \) was different?
3. Can you describe the graph of \( f(x) = x^2 + 6x + 5 \)?
4. What is the vertex of the parabola described by \( f(x) \)?
5. How does the value of \( f(x) \) change as \( x \) increases?

**Tip:** When substituting values into a function, always simplify carefully to avoid calculation mistakes.

Evaluate the function f(x) = x^2 + 6x + 5 at the given values of the independent variable and simplify:

a. f(8)
b. f(x + 5)
c. f(-x)

This page explains how to evaluate the quadratic function f(x) = x^2 + 6x + 5 at specific values, including f(8), f(x + 5), and f(-x). Suitable for students in Grades 9-11, the problem introduces key algebraic concepts such as function substitution and simplification.

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