Answer: C. $272$

Explanation:1. Understand the Properties of a Quadratic Function: A quadratic function can be written in vertex form as $f(x) = A(x-h)^2 + k$, where ($h,k$) is the vertex.

2. Use the Given Vertex: The vertex is given as ($1,6$). So, $h=1$ and $k=6$. The function becomes $f(x) = A(x-1)^2 + 6$.

3. Use one of the Given Points to Find A: The graph passes through the point ($2, 40$). Substitute $x=2$ and $f(x)=40$ into the equation: $40 = A(2-1)^2 + 6$ $40 = A(1)^2 + 6$ $40 = A(1) + 6$ $40 = A + 6$ $A = 40 - 6$ $A = 34$

(We can verify with the other point ($-1, 142$): $142 = 34(-1-1)^2 + 6$ $142 = 34(-2)^2 + 6$ $142 = 34(4) + 6$ $142 = 136 + 6$ $142 = 142$. This confirms our value of A is correct.)

4. Write the Specific Quadratic Function: With $A=34$, $h=1$, and $k=6$, the function is: $f(x) = 34(x-1)^2 + 6$

5. Calculate $f(-2)$: $f(-2) = 34(-2-1)^2 + 6$ $f(-2) = 34(-3)^2 + 6$ $f(-2) = 34(9) + 6$ $f(-2) = 306 + 6$ $f(-2) = 312$

6. Calculate $f(0)$: $f(0) = 34(0-1)^2 + 6$ $f(0) = 34(-1)^2 + 6$ $f(0) = 34(1) + 6$ $f(0) = 34 + 6$ $f(0) = 40$

7. Calculate $f(-2) - f(0)$: $f(-2) - f(0) = 312 - 40$ $f(-2) - f(0) = 272$

8. Conclusion: The value of $f(-2) - f(0)$ is $272$. This matches option C.