Factor $(a+b)(b+c)(c+a)+abc$. I know this is a cyclic polynomial, but I don't know how to solve problems like this. What should I do?
The polynomial $(a+b)(b+c)(c+a)+abc$ is not only cyclic, but also symmetric in $a,b,c$ and has total degree $3$, so if it factors, it should have a linear factor with total degree $1$. Also, it is "likely" that the linear factor is also cyclic and/or symmetric. So, try something like $a+b+c$ (the simplest symmetric polynomial in $a,b,c$ with total degree $1$) as one of the factors. Then determine what the other factor should be.
Spoiler: $(a+b)(b+c)(c+a)+abc = (a+b+c)(ab+bc+ca)$