To solve the given problem, we need to determine the formula for the average rate of change of $f(x) = -2x^2 - 5x + 3$ between $x = a$ and $x = a + 1$. The formula for the average rate of change is:

$\text{Average Rate of Change} = \frac{f(a+1) - f(a)}{(a+1) - a}$

This simplifies to:

$\text{Average Rate of Change} = f(a+1) - f(a)$

Step 1: Compute $f(a+1)$

Substitute $x = a+1$ into $f(x) = -2x^2 - 5x + 3$:

$f(a+1) = -2(a+1)^2 - 5(a+1) + 3$

Expand $(a+1)^2$:

$(a+1)^2 = a^2 + 2a + 1$

So:

$f(a+1) = -2(a^2 + 2a + 1) - 5(a+1) + 3$

Expand further:

$f(a+1) = -2a^2 - 4a - 2 - 5a - 5 + 3$

Combine like terms:

$f(a+1) = -2a^2 - 9a - 4$

Step 2: Compute $f(a)$

Substitute $x = a$ into $f(x) = -2x^2 - 5x + 3$:

$f(a) = -2a^2 - 5a + 3$

Step 3: Find $f(a+1) - f(a)$

Subtract $f(a)$ from $f(a+1)$:

$f(a+1) - f(a) = (-2a^2 - 9a - 4) - (-2a^2 - 5a + 3)$

Simplify the subtraction:

$f(a+1) - f(a) = -2a^2 - 9a - 4 + 2a^2 + 5a - 3$

Combine like terms:

$f(a+1) - f(a) = -4a - 7$

Step 4: Final Answer

The average rate of change is:

$f(a+1) - f(a) = -4a - 7$

Thus, the correct option is:

$\boxed{\text{A) } y = -4a - 7}$

Would you like further clarification or details? Here are 5 related questions to deepen your understanding:

1. What is the general formula for the average rate of change of a quadratic function?
2. How would the process differ if $f(x)$ were a linear function?
3. Why does the denominator in the average rate of change formula cancel in this case?
4. How does the second derivative of $f(x)$ relate to the rate of change of the function?
5. Can you verify the solution by graphing $f(x)$ and estimating the slope between $x = a$ and $x = a+1$?

Tip: Always expand and simplify carefully when working with quadratic functions to avoid errors in subtraction or distribution.