The sum of all possible solutions is \(\mathbf{5}\), so the correct answer is \(\mathbf{D}\).


Briefly:
Start with

$$
\(\sqrt{2 x^2-x-9}=x+1\)
$$

1. Since the square root is nonnegative, we need \(x+1 \geq 0 \Rightarrow x \geq-1\).
2. Square both sides:

$$
\(2 x^2-x-9=(x+1)^2=x^2+2 x+1\) .
$$

3. Rearrange:

$$
\(2 x^2-x-9-x^2-2 x-1=0 \Rightarrow x^2-3 x-10=0\) .
$$

4. Factor:

$$
\((x-5)(x+2)=0 \Rightarrow x=5 \text { or } x=-2\)
$$

5. Check the domain condition \(x \geq-1\) : only \(x=5\) is valid.



So the only solution is \(x=5\), and the sum of all solutions is 5 .