The expression you've highlighted in red is a bit nasty, but truly it's no different from any other expression you've expanded in the past. My best guess is that you're getting intimidated and can't figure it out because you're taking in the whole thing at once and it's just too much. To counteract this, let's break it down and analyze it part-by-part, performing just one step at a time. First, write down the unsimplified version, just so we know what we're dealing with. I've separated out the nested grouping symbols instead of using only parentheses, and inserted the implied multiplication symbols:
\(\displaystyle \dfrac{(k+1) \cdot [(k+1)+1] \cdot \{[2(k+1)]+1\}}{6}\)
Now, the first term seems okay on its own. I see no immediate simplification/expansion to be done, so let's isolate the middle term and see what we can do with it:
\(\displaystyle [(k+1)+1]\)
Looking at it, it should be obvious why the redundant parentheses can be removed simplifies down to k + 2. Okay, so what about the final term? What can we do there?
\(\displaystyle \{[2(k+1)]+1\}\)
Do you see that we can expand the 2(k+1) part into 2k + 2, thus making the whole term into 2k + 3? It really seems like we're getting somewhere! Since we've simplified/expanded each individual term, let's put everything back in and see where we stand now:
\(\displaystyle \dfrac{(k+1) \cdot (k+2) \cdot (2k+3)}{6}\)
Try continuing from here by multiplying all the terms together. What do you get? How does that relate to the expression highlighted in green?