Vieta's take too long

Algebra Level 5

\[100 (a_1^2 + 1)(a_2^2 + 1)\cdots (a_{10}^2 + 1) \]

Let \(a_1, a_2, \ldots,a_{10} \) denote the roots to the equation \( \displaystyle \sum_{m=1}^{10} m x^m = 0 \). Find the value of the expression above.