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------------------------------------------------ Evaluate the function [tex]f(x) = x^2 + 2x + 9[/tex] at the given values of the independent variable and simplify.

a. [tex]f(8)[/tex]

b. [tex]f(x+2)[/tex]

a. [tex]f(8) = 89[/tex] (Simplify your answer.)

b. [tex]f(x + 2) = \square[/tex] (Simplify your answer.)

To evaluate the function [tex]$ f(x) = x^2 + 2x + 9 $[/tex] at the given values, follow these steps:

### Part a: Evaluate [tex]$ f(8) $[/tex]

1. Substitute [tex]$ x = 8 $[/tex] into the function:
Plug [tex]$ 8 $[/tex] into [tex]$ f(x) = x^2 + 2x + 9 $[/tex].

[tex]\[
f(8) = 8^2 + 2 \times 8 + 9
\][/tex]

2. Calculate each term:
- [tex]$ 8^2 = 64 $[/tex]
- [tex]$ 2 \times 8 = 16 $[/tex]

3. Add the values together:
[tex]\[
f(8) = 64 + 16 + 9 = 89
\][/tex]

So, the simplified value is [tex]$ f(8) = 89 $[/tex].

### Part b: Evaluate [tex]$ f(x+2) $[/tex]

1. Substitute [tex]$ x = x+2 $[/tex] into the function:
Replace [tex]$ x $[/tex] with [tex]$ (x+2) $[/tex] in [tex]$ f(x) = x^2 + 2x + 9 $[/tex].

[tex]\[
f(x+2) = (x+2)^2 + 2(x+2) + 9
\][/tex]

2. Expand each term:
- [tex]$ (x+2)^2 = x^2 + 4x + 4 $[/tex]
- [tex]$ 2(x+2) = 2x + 4 $[/tex]

3. Combine all terms:
[tex]\[
f(x+2) = x^2 + 4x + 4 + 2x + 4 + 9
\][/tex]

4. Simplify by combining like terms:
- Combine the [tex]$ x $[/tex] terms: [tex]$ 4x + 2x = 6x $[/tex]
- Combine the constant terms: [tex]$ 4 + 4 + 9 = 17 $[/tex]

So, [tex]$ f(x+2) = x^2 + 6x + 17 $[/tex].

Therefore, the simplified expression for [tex]$ f(x+2) $[/tex] is [tex]$ x^2 + 6x + 17 $[/tex].