Inverse Z transform of a left sided signal
Note that the order of your numerator equals the denominator, first perform a long division to simplify, before performing partial fraction expansion:

$$ H(z) = \frac{ 1 + 1/6 z^{-1}}{1 - 0.25 z^{-1}} = -\frac{2}{3} + \frac{5/3}{1 - 0.25 z^{-1}} $$

(Note: long divison already simplified the expression that it does not require a further partial fraction expansion to be applied, hence skip step-2 now)

Now apply the inzerve Z-transform and note that for left sided inverse sequence wil be $-a^n u[-n-1]$ for a term like $1/(1-az^{-1})$.