In the marking scheme they somehow manipulated a cubic to retrieve one of the factors needed to answer the question:
My question is: How can it be known to do this baring in mind there is three roots and the others had many decimals? Its like they pulled it from nowhere.
Let $p(k)=k^{3}+3k^{2}-36k+52$. Since $p(2)=0$, the Factor Theorem
gives that $k-2$ is a factor of $p(k)$.
You then divide the polynomial $p(k)$ by $k-2$ to obtain the quotient $k^{2}+5k-26$. Thus, $p(k)=(k-2)(k^{2}+5k-26)$. This is the correct factorization.