To solve for the value of $a$, let’s examine the given equation:

$\frac{24x^2 + 25x - 47}{ax - 2} = -8x - 3 - \frac{53}{ax - 2}$

Since this equation is true for all values of $x$ (except $x = \frac{2}{a}$, where the denominator would be zero), we can clear the denominators by multiplying both sides by $ax - 2$. This gives:

$24x^2 + 25x - 47 = (-8x - 3)(ax - 2) - 53$

Now, expand the right side:

$(-8x - 3)(ax - 2) = -8ax^2 + 16x - 3ax + 6$

Thus, the equation becomes:

$24x^2 + 25x - 47 = -8ax^2 + 16x - 3ax + 6 - 53$

Simplifying the constants on the right side:

$24x^2 + 25x - 47 = -8ax^2 + 16x - 3ax - 47$

Now, we compare coefficients for each power of $x$ on both sides of the equation:

1. For $x^2$ term: $24 = -8a$

   Solving for $a$:

   $a = -\frac{24}{8} = -3$

So, the value of $a$ is $\boxed{-3}$.

Would you like further clarification on any part of this solution, or do you have additional questions?

Here are five related questions:

1. How would the solution change if $a$ had a different sign?
2. Can this approach be applied to equations with higher-degree polynomials?
3. How do we handle cases where $x = \frac{2}{a}$?
4. What if the constants on the right side differed; how would that affect the result?
5. How do we verify if our solution for $a$ satisfies the equation?

Tip: When solving polynomial equations, comparing coefficients after clearing denominators is a powerful technique to isolate variables.