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.