How to apply the Chain rule when it involves a quotient with two terms multiplied in the denom. and num.?

use the quotient rule first and then the other rules...
answer; 4x cos(x)/(sqrt(x-1)(3x-1))-(2(3x^2+x-2) sin(x)/((x-1)^(3/2) (3x-1))

I think it would be a lot easier if you rewrite the equation:
y=
(4x)(sin(x))*((3x-1)^-1)*((x-1)^-1/2)
and then use the product rule.  I always find
quotient rule is harder to remember...

Yes start withthe quotient rule.  You really don't need to use the chain rule here.  

The chain rule is needed when dealing with functions of one or more variables which are themselves functions of other variables, such as e^sin(x) where f(x) = e^x and g(x)=sin(x) and you want to find f'(g(x)).

Here you can just substitute

u=4x, v=sin(x) w=(3x-1) z=?(x-1)

y=uv/(wz)

Since every function is of the same variable the chain rule doesn't apply.  Apply the quotient rule instead:

y' = [(wz) (uv)' - (uv)(wz)']/(wz)²

Now apply the product rule

(uv)' = u'v + uv' and (wz)' = w'z + wz'

y' = [(wz) (u'v + uv') - (uv)(w'z + wz')]/(wz)²

Susbstitute the following back in

u=4x, v=sin(x) w=(3x-1) z=?(x-1)
u'=4, v'=cos(x), w'=3, z'=1/[2?(x-1)]

y' = [(3x-1)?(x-1) (4sin(x) + 4cos(x)) -
...... (4xsin(x))(3?(x-1) + (3x-1)/(2?(x-1))]/[(3x-1)?(x-1)]²

It's quite ugly.  Just use algebra to simplify it.