Explanation of a Solution to a Mathematical Induction problem.

Question

I'm quite stuck on the explanation that my university has provided for a tutorial question.

Here is the question with their working.

I'm just wondering how they got from line 6 to line 7.

Answer 1

Considering line 6 is $$\sum_{i=1}^{k}(3i-1)^2 = \frac{1}{2} \cdot(6(k^3+3k^2+3k+1)+3k^2+5k+2) $$ We want to factorize this line in order to find which term could be factorized by (k+1). We know that $(a+b)^3 = a^3 + 3ba^2+3b^2a+b^3$ and that $(a+b)^2 = a^2 + 2ab + b^2$.
The first step is to recognize $(k^3+3k^2+3k+1) = (k+1)^3$ and we're left with $3k^2+5k+2 = 3k^2 + 6k +3 - 1 -k = 3(k+1)^2 - (1+k)$.
You can rewrite it as $$\sum_{i=1}^{k}(3i-1)^2 = \frac{1}{2}\cdot(6(k+1)^3 + 3(k+1)^2 - (1+k))$$ and then factorize by $(k+1)$ which give you the answer you were looking for.