Please look at attached images to understand the question first.
can some one explain or elaborate how
$λ^3 - 8 λ^2 + 17λ - 4$ becomes
$(λ-4) (λ^2-4λ+1)$
and how do we find these values of λ : $ λ=4, λ=2 + \sqrt 3, λ= 2 - \sqrt 3
$
look here for example which i am referring to 
The eigenvalues of $A$ are the roots of $λ^3−8λ^2+17λ−4$.
As stated in the image you provide (it looks like a math book, so that property is most likely demonstrated prior to being used), the polynomial $λ^3−8λ^2+17λ−4$ has roots $\pm1$, $\pm2$, or $\pm4$. You can determine which ones are roots and which ones aren't through trial and error.
By using polynomial long division, you can determine that $4$ is a root, and determine that the remainder of the division of $(λ^3−8λ^2+17λ−4)$ by $(\lambda -4)$ is $(\lambda^2-4\lambda+1)$. You can verify this result by expanding $(\lambda -4)(\lambda^2-4\lambda+1)$.
The roots of the second degree polynomial $(\lambda^2-4\lambda+1)$ are $2+\sqrt 3$ and $2-\sqrt 3$.
As a result, the eigenvalues of $A$ are $\{ 4, 2+\sqrt 3, 2-\sqrt 3\}$.