Which example correctly demonstrates polynomial addition?

To determine which example correctly demonstrates polynomial addition, we should add the two given polynomials in each option.

Let's add the polynomials:

\[
(2x^2 + 6x + 1) + (3x^2 + 3x + 9)
\]

1. Combine the \(x^2\) terms:
\[
2x^2 + 3x^2 = 5x^2
\]

2. Combine the \(x\) terms:
\[
6x + 3x = 9x
\]

3. Combine the constant terms:
\[
1 + 9 = 10
\]

So the correct addition results in:
\[
5x^2 + 9x + 10
\]

Now, let's examine the options:

A. \((2x^2 + 6x + 1) + (3x^2 + 3x + 9) = 5x^2 + 9x + 10\) (Correct)

B. \((2x^2 + 6x + 1) + (3x^2 + 3x + 9) = 5x^4 + 9x^2 + 10\) (Incorrect, exponent of \(x\) is wrong)

C. \((2x^2 + 6x + 1) + (3x^2 + 3x + 9) = -x^2 + 3x - 8\) (Incorrect, all terms are wrong)

D. \((2x^2 + 6x + 1) + (3x^2 + 3x + 9) = 13x^2 + 9x + 10\) (Incorrect, \(x^2\) terms are added incorrectly)

Only option A is correct.

So the answer is:
**. (2x^2 + 6x + 1) + (3x^2 + 3x + 9) = 5x^2 + 9x + 10