To simplify this expression, we first need to expand the cube of the binomial $3b^{2}c$^4 using the power of a power rule:

$3b^{2}c$^4 = 3^4 $b^2$^4 c^4
= 81 b^8 c^4

Now, we substitute this back into the original expression:

-81b^6c^3 / $3b^{2}c$^4
= -81b^6c^3 / $81b^8{c}^4$

Now, we can simplify this expression by dividing each term in the numerator by the corresponding term in the denominator:

= -1/b^2

Therefore, the simplified expression is -1/b^2.