I want to divide the following polynomial (in terms of $t$) with coefficients in terms of $\lambda$.

$$(\lambda^6 - 5\lambda^4 + 6\lambda^2 - 1)t^5 + (\lambda^5 - 4\lambda^3 + 3\lambda^2)t^6$$ 

by $$ \lambda t^2 -\lambda^2 t + \lambda$$

The resulting quotient will include a fractional component (the numerator's
degree will be strictly less than the denominator's degree -- is there any way to find the partial fraction decomposition of this expression, which will be in terms of $\lambda$).

EDIT:

I have not yet finished the hand-written calculation above, but this is what it might like:

$$ t(\frac{2\lambda^8 - 9 \lambda^6 + 2 \lambda^5 + 6 \lambda^4 - 4\lambda^2}{\lambda}) + t^3(\frac{2\lambda^6 - 9\lambda^4 + 3\lambda^3 + 6\lambda^2 -1 }{\lambda}) + \frac{t(\lambda^4 - 2\lambda) + (\lambda^3 - 4\lambda^2)}{\lambda t^2 - \lambda^2 t + \lambda}$$