Which example correctly demonstrates polynomial addition? (1 point)

To check which example correctly demonstrates polynomial addition, we need to add the two given polynomials:

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

Now, let's add the like terms:

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:**
\[
1 + 9 = 10.
\]

Putting it all together, we get:

\[
(2x^2 + 6x + 1) + (3x^2 + 3x + 9) = 5x^2 + 9x + 10.
\]

Now let's go through the provided options:

1. \((2x^{2}+6x+1)+(3x^{2}+3x+9)=13x^{2}+9x+10\) **(incorrect)**
2. \((2x^{2}+6x+1)+(3x^{2}+3x+9)=5x^{4}+9x^{2}+10\) **(incorrect)**
3. \((2x^{2}+6x+1)+(3x^{2}+3x+9)=-x^{2}+3x-8\) **(incorrect)**
4. \((2x^{2}+6x+1)+(3x^{2}+3x+9)=5x^{2}+9x+10\) **(correct)**

Thus, the correct example that demonstrates polynomial addition is:

\[
\boxed{(2x^{2}+6x+1)+(3x^{2}+3x+9)=5x^{2}+9x+10}.
\]