I am studying the growth rate of a weighted variant of self-avoiding walks and came up with a linear recurrence in terms of the fixed weights $a$ and $b$ whose characteristic polynomial contains the factor $$ f(x) = a^3b^3 + \left(a^3b^2+ a^2b^3\right)x + 2a^2b^2 x^2 - \left(a^2b + ab^2\right)x^3 - 2abx^4 -(a + b)x^5 + x^6. $$ The goal is to extract the largest root as a function of $a$ and $b$ since this value is an upper bound on the exponential growth rate. I used Mathematica to get to this point, so my question is whether or not there is any hope to extract a couple more roots of $f(x)$ in order to find all of them. The polynomial seems well-structured and I'm assuming the constants $a$ and $b$ are positive.
For context, a more naive recurrence leads to a characteristic equation whose largest root is $$ \frac{a+b+\sqrt{a^2+14ab+b^2}}{2}, $$ and setting $a=b=1$ recovers the trivial $3^n$ upper bound given by non-backtracking walks.
It might help to note that we can remove one degree of freedom by writing $b=ac,\; x=a t$: your polynomial becomes $$ a^6 ({t}^{6} - \left( c+1 \right) {t}^{5}-2\,c{t}^{4} - \left( {c}^{2}+c \right) {t}^{3}+2\,{c}^{2}{t}^{2}+ \left( {c}^{3}+{c}^{2} \right) t+{ c}^{3}) = a^6 P(t)$$
Here is a plot of the real roots of $P(t)$ as a function of $c$ ($t$ on the horizontal axis, $c$ on the vertical):
Here is a closeup of the region near $(0,0)$:
The discriminant of $P(t)$ with respect to $t$ is
$$ -400\,{c}^{20}-2480\,{c}^{19}+101696\,{c}^{18}-22608\,{c}^{17}+152352 \,{c}^{16}+116576\,{c}^{15}+152352\,{c}^{14}-22608\,{c}^{13}+101696\,{ c}^{12}-2480\,{c}^{11}-400\,{c}^{10} $$
The number of real roots can change at the roots of the discriminant, approximately $$-19.46223637, -0.05138155663, 0, 0.07643350517, 13.08326758$$
Thus for $c > 13.083\ldots$ there are $4$ real roots, while for $0.0764\ldots < c < 13.083\ldots$ there are $2$, and for $0 < c < 0.0764\ldots$ there are again $4$.
If I'm not mistaken, as $c \to +\infty$ the largest root is asymptotically
$$ c + 4-18\,{c}^{-1}+182\,{c}^{-2}-2244\,{c}^{-3}+30786\,{c}^{-4}-451334\,{c }^{-5}+O \left( {c}^{-6} \right) $$
while the second is
$$ \frac{\sqrt{2\sqrt{5}-2}}{2} c^{1/2} + \frac{\sqrt{5}-1}{2} - \frac{7 \sqrt{2\sqrt{5}-2}(5 - \sqrt{5})}{40} c^{-1/2} + \frac{25 - 9 \sqrt{5}}{5} c^{-1} + O(c^{-3/2}) $$