To solve for $(f+g)(x)$, we need to add the functions $f(x)$ and $g(x)$.

Given:

$f(x) = 2x^2 - 2x + 7 \quad \text{and} \quad g(x) = 4x^2 - 2x + 9$

Step 1: Add the functions

We sum up corresponding terms of $f(x)$ and $g(x)$: $(f+g)(x) = (2x^2 - 2x + 7) + (4x^2 - 2x + 9)$

Combine like terms:

1. Combine $x^2$-terms: $2x^2 + 4x^2 = 6x^2$
2. Combine $x$-terms: $-2x - 2x = -4x$
3. Combine constants: $7 + 9 = 16$

Thus: $(f+g)(x) = 6x^2 - 4x + 16$

Step 2: Identify the correct option

Looking at the options:

• Option D: $(f+g)(x) = 6x^2 - 4x + 16$

Final Answer:

The correct answer is D.

Would you like me to explain further or clarify any step?
Here are 5 related questions you might explore:

1. How do you subtract two functions $f(x)$ and $g(x)$?
2. What happens when you multiply two polynomials instead of adding them?
3. How can you graph the resulting function $(f+g)(x)$?
4. How do you determine the degree of a sum of polynomials?
5. What are the steps to simplify polynomials involving higher powers?

Tip: Always group like terms carefully when adding or subtracting polynomials.