The sign of a quadratic on its root interval

If $x > 0$ and $y < 0$, prove that $y^2 + y - 2 \leq 0$ for all $y$ in $[-2, 1]$.

Source: InternLM Lean-Workbook, record lean_workbook_53513 (Apache-2.0). The complete source proposition and explicit variable declarations are preserved.