Step 1:

The combined equation of the asymptotes is $(2x+3y-8)(3x-2y+1)=0$

The equation of the hyperbola differs only in the constant term.

Let the equation be $(2x+3y-8)(3x-2y+1)=k$

Step 2:

The hyperbola passes through $(5,3)$

Therefore $(10+9-8)(15-6+1)=k$

$\Rightarrow k=11\times 10=110$

The equation is $(2x+3y-8)(3x-2y+1)=110$

On simplifying we get,

$6x^2+5xy-6y^2-22x+19y+8=110$

$\Rightarrow 6x^2+5xy-6y^2-22x+19y-102=0$

This is the required equation .