\left(6x^{4}-5x^{3}+3x-8x\right)-\left(4x^{4}+5x^{3}+3x^{2}+12x\right)
To subtract these polynomials, combine like terms.
Starting with the terms of degree 4, we have $6x^4 - 4x^4 = \boxed{2x^4}$.
Next, looking at the terms of degree 3, we have $-5x^3 - 5x^3 = \boxed{-10x^3}$.
Moving on to the terms of degree 2, we have $0x^2 - 3x^2 = \boxed{-3x^2}$.
Finally, for the terms of degree 1, we have $3x - 8x - 12x = \boxed{-17x}$.
There are no terms of degree 0, so we don't need to include a constant term.
Putting it all together, we have $\boxed{2x^4 - 10x^3 - 3x^2 - 17x}$.