The equation 24 x 2 + 25 x − 47 a x − 2

To solve for $a$, we can analyze the given equation:

$24x^2 + 25x - 47a \cdot (x - 2) = -8x - 3 - 53a \cdot (x - 2)$

This equation is stated to be true for all $x \neq 2a$, which suggests that the terms on both sides involving $x$ must have matching coefficients.

Step 1: Expand and simplify both sides

First, let's expand both sides:

1. Left Side: $24x^2 + 25x - 47a \cdot x + 47a \cdot 2$ Simplifies to: $24x^2 + 25x - 47a \cdot x + 94a$

2. Right Side: $-8x - 3 - 53a \cdot x + 53a \cdot 2$ Simplifies to: $-8x - 3 - 53a \cdot x + 106a$

Step 2: Match coefficients of $x^2$, $x$, and constants

For this equation to hold for all values of $x \neq 2a$, the coefficients of corresponding powers of $x$ on both sides must be equal.

Matching $x$ terms:

The terms involving $x$ are: $-47a \text{ on the left side, and } -8 - 53a \text{ on the right side.}$ Setting these equal gives: $-47a = -8 - 53a$

Rearrange to isolate $a$: $53a - 47a = -8$ $6a = -8$ $a = -\frac{8}{6} = -\frac{4}{3}$

Conclusion:

There is no answer among the provided choices that matches $-\frac{4}{3}$. This indicates either a misinterpretation of the problem or a potential error in the given answer choices. If needed, we can revisit the problem with more information.