How to show the polynomial $P= x^{10}+ax^9+bx^8+cx^7+x+1$ always has at least one non-real root? I rephrase my question how can prove or disprove that the roots of polynomial $P= x^{10}+ax^9+bx^8+cx^7+x+1$ can't be all real any way we choose $a,b, c\in \mathbb{R}$.

As Pete commented, Descartes' rule of signs tells us that $P$ has at most four positive roots no matter how $a,b,c$ are chosen, and at most four negative roots no matter how $a,b,c$ are chosen. Therefore it must have at least two complex roots. (In fact it must have at least four complex roots, as it happens, since $a,b,c$ can't be chosen to simultaneously give four positive and four negative roots.) And an examination of a proof of the rule reveals that it does take multiple roots into account.
