lean_workbook_plus_64393

If $a,b,c$ are real numbers such that $a+b+c=0$ , then prove that \n $\cfrac{a^5+b^5+c^5}{5}=\left(\cfrac{a^3+b^3+c^3}{3}\right)\left(\cfrac{a^2+b^2+c^2}{2}\right)$ \nThis is Problem 33 on pg. 532 of Pre-College Math. \n\nRather if there is any elegant way and some clever trick that may be applied is preferable, for even I have a solution by just substituting for $c$ as $-a-b$ and then expanding everything on the LHS and RHS.