Estimation of a polynomial I'm currently reading the following paper: http://arxiv.org/abs/1209.0612 and got stuck on Proposition 3.1 (2). 
The claim translated to polynomials is the following:

Assume $n\geq 3, c\geq 1, d\geq 1$ are natural numbers such that $c²+d²-(n-1)cd<0$. Show that $(n³-n+1)c²+(n+1)d²-(n²+n-1)cd>1$.

Anyone an idea to solve this?

Let $c,d$ be positive real numbers and $n>1$ (esp., $n^3-n+1>0$).
By the arithmetic-geometric inequality
$$(n^3-n+1)c^2+(n+1)d^2\ge 2\cdot\sqrt{(n^3-n+1)(n+1)}\cdot c d.$$
One checks by multiplying out that
$$4(n^3-n+1)(n+1)=(n^2+n-1)^2+3+3n^2(n^2-1)+2n(n^2+1),$$ 
hence $2\cdot\sqrt{(n^3-n+1)(n+1)}>n^2+n-1$ and finally
$$(n^3-n+1)c^2+(n+1)d^2>(n^2+n-1) cd. $$
