problem to determine the chromatic polynomial of a graph

for a homework graph theory, I'm asked to determine the chromatic polynomial of the following graph

For the Descomposition Theorem of Chromatic Polynomials. if G=(V,E), is a connected graph and e belong E

P (G, λ) = P (Ge, λ) -P(Ge', λ)

where Ge denotes de subgraph obtained by deleting de edge e from G (Ge= G-e) and Ge' is the subgraph obtained by identifying the vertices {a,b} = e

When calculating chromatic Polynomials, i shall place brackets about a graph to indicate its chromatic polynomial. removes an edge any of the original graph to calculate the chromatic polynomial by the method of decomposition.

 P (G, λ) = P (Ge, λ)-P (Ge', λ) = λ (λ-1)^4 - [λ(λ-1)*(λ^2 - 3λ + 3)]

But the response from the answer key and the teacher is:

P (G, λ) = λ (λ-1)(λ-2)(λ^2-2λ-2)

I have operated on the polynomial but I can not reach the solution that I ask .. what am I doing wrong?

math.stackexchange.com told me as a way to solve my problem. Here's the solution:

https://math.stackexchange.com/questions/33946/problem-to-determine-the-chromatic-polynomial-of-a-graph