How to simplify (x-3)/(2x) - (4x + 3)/4?

the LCD is 4x. So, that gives you

2(x-3)/(4x) - x(4x+3)/(4x)
= [2(x-3)-x(4x+3)]/(4x)
= (2x-6-4x^2-3x)/(4x)
= -(4x^2+x+6)/(4x)

That's about as far as you can take it.