Okay, let's tackle this problem step by step. The question is asking for the dot product of two vectors. Hmm, I remember the dot product involves multiplying corresponding components of the vectors and then adding them up. Let me make sure I recall the exact formula. 

So, if we have two vectors, say **a** = [a₁, a₂, a₃] and **b** = [b₁, b₂, b₃], their dot product is a₁*b₁ + a₂*b₂ + a₃*b₃. Right? Yeah, that sounds right. So I need to apply this formula to the given vectors.

The first vector is (-2, 0, 7) and the second one is (3, 4, -5). Let me write them down again to avoid mixing up the components. 

First component: -2 and 3. Multiply those: (-2)*3 = -6. 

Second component: 0 and 4. Multiplying those gives 0*4 = 0. 

Third component: 7 and -5. Multiplying those: 7*(-5) = -35. 

Now, add all those products together: -6 + 0 + (-35). Let me compute that. 

-6 plus 0 is -6. Then adding -35 to that gives -6 -35 = -41. 

Wait, let me double-check my calculations to be sure. 

First term: -2 * 3. Negative times positive is negative. 2*3=6, so -6. Correct. 

Second term: 0 multiplied by anything is 0. So 0*4=0. Correct. 

Third term: 7*-5. Positive times negative is negative. 7*5=35, so -35. Correct. 

Adding them up: -6 + 0 = -6. Then -6 + (-35) is indeed -41. 

Hmm, seems straightforward. But let me confirm once more. Maybe I made a mistake in signs? 

First vector is (-2, 0, 7). Second is (3, 4, -5). 

First multiplication: -2*3 = -6. Correct. 

Second: 0*4 = 0. Correct. 

Third: 7*(-5) = -35. Correct. 

Sum: -6 + 0 + (-35) = -41. Yep, that's right. 

I think that's the answer. But just to be thorough, maybe there's another way to compute the dot product? Let me recall. The dot product is also equal to the product of the magnitudes of the vectors multiplied by the cosine of the angle between them. But since we don't have the angles here, that method isn't helpful in this case. So the component-wise multiplication and sum is the way to go. 

Alternatively, if I were to write this out in LaTeX, it would be:

\[
\begin{pmatrix} -2 \\ 0 \\ 7 \end{pmatrix} \cdot \begin{pmatrix} 3 \\ 4 \\ -5 \end{pmatrix} = (-2)(3) + (0)(4) + (7)(-5) = -6 + 0 -35 = -41
\]

Yes, that's exactly what I did. So the final answer should be -41. 

I don't see any mistakes in my calculations. All steps check out. The signs are correct, and the arithmetic is straightforward. Therefore, I'm confident that the dot product of the two vectors is -41.
Therefore, the dot product of $\begin{pmatrix} -2 \\ 0 \\ 7 \end{pmatrix}$ and $\begin{pmatrix} 3 \\ 4 \\ -5 \end{pmatrix}$ is $\boxed{-41}$.
</think>