Value of $a$ ?

$\begin{aligned} \frac {24x^2+25x-47}{ax-2} & = -8x-3-\frac {53}{ax-2} \\ & = \frac {-(8x+3)(ax-2)-53}{ax-2} \\ & = \frac {-(8ax^2+(3a-16)x-6)-53}{ax-2} \\ & = \frac {-8ax^2 +(16-3a)x-47}{ax-2} \end{aligned}$

Equating coefficients of $x^2$ and $x$ on both sides $\implies a=\boxed{-3}$ .

Relevant wiki: FOIL Method

The faster way is to multiply each side of the given equation by ax−2 (so you can get rid of the fraction). When you multiply each side by ax−2, you should have:

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

You should then multiply (−8x−3) and (ax−2) using FOIL.

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

Then, reduce on the right side of the equation

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

Since the coefficients of the $x^2$ -term have to be equal on both sides of the equation, $-8a = 24,$ or $a = -3.$ (Answer)