The polynomial $x^{2n} - 2x^{2n-1} + 3x^{2n-2}- \cdots -2nx + 2n + 1$ has no real roots. 
Prove that the polynomial $$x^{2n} - 2x^{2n-1} + 3x^{2n-2}- \cdots -2nx + 2n + 1$$ has no real roots.

Here is the solution
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*For $x \leq 0$ we have obviously $p(x)>0$. Let $x>0$. We transform the polynomial in the same way as a geometric series:
  $$
\begin{aligned}
p(x) &=x^{2 n}-2 x^{2 n-1}+3 x^{2 n-2}-\cdots-2 n x+2 n+1 \\
x p(x) &=x^{2 n+1}-2 x^{2 n}+3 x^{2 n-1}-4 x^{2 n-2}+\cdots+(2 n+1) x.
\end{aligned}
$$
  Adding, we get
  $$
\begin{aligned}
x p(x)+p(x) &=x^{2 n+1}-x^{2 n}+x^{2 n-1}-x^{2 n-2}+\cdots,+x+2 n+1 \\
(1+x) p(x) &=x \cdot \frac{1+x^{2 n+1}}{1+x}+2 n+1.
\end{aligned}
$$
  From here we see that $p(x)>0$ for $x>0$.
  

Can anyone explain how the last line of the solution implies that the polynomial has no real roots?

Solution shows that $p(x)>0, \text{ for all } x$, which means that there is no any value of $x$ such that $p(x)=0$ 
