Is right this chromatic polynomial for this Bridge Graph? I have the following graph:
Bridge Graph with N = 8 
And I need to find its chromatic polynomial.
Based in my notes, I have reached the following result:
$$Pg(x) = \frac{((x-1)^4 + (x-1))^2}{x(x-1)} $$.
It is because I have applied this formula:
$$Pg(x) = \frac{Pg1(x) Pg2(x)}{Pkr(x)} $$.
Taking the Bridge Graph and the previous formula, I have replaced with:
$$ Pg1(x) = Pg2(x) = Pc4(x)= (x-1)^4+(x-1) $$
Because both sub-graphs are cycles. 
On the other hand, the Pkr(x) in this case is equals to  Pk2(x). So, the chromatic polynomial for a complete graph with n = 2, could be expressed as :
$$x(x-1)$$
So, my main doubt is if this reasoning is right. Because I'm not sure about my resolution and I think that Ia have missing something in the final response.
Thanks a lot!

Using standard techniques:
$$x^8-9 x^7+36 x^6-82 x^5+114 x^4-96 x^3+45 x^2-9 x$$
