QPaperGen

We know a function is strictly increasing on an interval if its derivative is positive on that interval.

\[
f'(x) = \frac{d}{dx}(x^2 + ax + 1) = 2x + a
\]

We require:
\[
f'(x) > 0 \quad \text{for all } x \in (1, 2)
\]

This means:
\[
2x + a > 0 \quad \text{for all } x \in (1, 2)
\]

The minimum value of \( f'(x) = 2x + a \) in the interval \( (1, 2) \) occurs at the left endpoint \( x = 1 \).

\[
\text{So, we need: } 2(1) + a > 0 \Rightarrow 2 + a > 0 \Rightarrow a > -2
\]

Thus, the least value of \( a \) such that \( f(x) \) is strictly increasing on \( (1, 2) \) is:
\[
\boxed{-2}
\]