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Answer: 1/3 (7 – 2 √19) ≤ a < 2.

Algorithm.

You solve an equation with an arbitrary parameter a, and you get x₁ and x₂, no matter in what order.

Solve the inequalities x₁ < 2 and x₂ with respect to a

Compare the resulting intersection intervals.

P.S. If you use, for example, Wolfram Mathematica, the answer is obtained in two clicks, just enter this:

{x1, x2} = x /. Solve[(a – 2) x^2 – 2 (a + 3) x + 4 a == 0, x];

Reduce[x1 < 3 && x2 < 2]