Solution — 2021 Question 1 Quadratic Equations & Inequalities

For the quadratic to be always positive, the coefficient of $x^{2}$ must be positive (which is true) and the discriminant must be negative.

\begin{matrix} b^{2} - 4 a c & < 0 \\ {(k - 1)}^{2} - 4 (k + 2) (1) & < 0 \\ k^{2} - 2 k + 1 - 4 k - 8 & < 0 \\ k^{2} - 6 k - 7 & < 0 \\ (k + 1) (k - 7) & < 0 ∎ \end{matrix}

$- 1 < k < 7 ∎$