Some nasty calculations involving `incomplete' $\Gamma$ function

In some of my work I came to such problem. Let $n$ be a non-negative integer. Is there any elegant (or not?) way to show that $$2\left(1+\frac{n+1}{e}\right) - (n+2)(n+1)\left[ \frac 1e \int_0^1 x^n e^x dx + (2-e) \int_0^1 x^n e^{-x} dx \right] \neq 0$$ for all $n$'s? I am pretty sure this is true but have no idea how to prove it nicely.

If someone is interested, this expression is the determinant of a system of equations which I would like to show have a unique solution.

I know that $\int x^n e^x dx = e^x \sum_{k=0}^n (-1)^k \frac{n! x^{n-k}}{(n-k)!}$, but this does not help me.

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