Let's break down the given problem based on the image.

You are given two functions:

$f(x) = 2x^2 - 8x + 5$ $g(x) = x^2 + 16$

You are asked to find three different operations involving these functions:

a) $(f + g)(x)$

This is the sum of the two functions, $f(x) + g(x)$.

$(f + g)(x) = (2x^2 - 8x + 5) + (x^2 + 16)$

Simplify the expression:

$(f + g)(x) = 2x^2 + x^2 - 8x + 5 + 16$

$(f + g)(x) = 3x^2 - 8x + 21$

b) $(f - g)(x)$

This is the difference between the two functions, $f(x) - g(x)$.

$(f - g)(x) = (2x^2 - 8x + 5) - (x^2 + 16)$

Simplify the expression:

$(f - g)(x) = 2x^2 - x^2 - 8x + 5 - 16$

$(f - g)(x) = x^2 - 8x - 11$

c) $(f \cdot g)(x)$

This is the product of the two functions, $f(x) \cdot g(x)$.

$(f \cdot g)(x) = (2x^2 - 8x + 5)(x^2 + 16)$

Expand this by distributing each term:

$(f \cdot g)(x) = 2x^2(x^2 + 16) - 8x(x^2 + 16) + 5(x^2 + 16)$

Now, distribute each term:

$= 2x^4 + 32x^2 - 8x^3 - 128x + 5x^2 + 80$

Simplify the terms:

$(f \cdot g)(x) = 2x^4 - 8x^3 + 37x^2 - 128x + 80$

Final Answers:

• $(f + g)(x) = 3x^2 - 8x + 21$
• $(f - g)(x) = x^2 - 8x - 11$
• $(f \cdot g)(x) = 2x^4 - 8x^3 + 37x^2 - 128x + 80$

Would you like further details or explanations on any part of this? Here are five related questions you can try:

1. How do you find the quotient of two functions, $(f / g)(x)$?
2. What are the domain restrictions for each function $f(x)$ and $g(x)$?
3. How would you evaluate $(f + g)(2)$?
4. What are the critical points of $f(x)$ and $g(x)$?
5. How do you find the composition of the functions, such as $f(g(x))$?

Tip: When adding or subtracting functions, remember to combine like terms to simplify the expression properly.