By the "quadratic formula" the roots of \(\displaystyle 2qx^2 + 5qx + 5q - 3 = 0\) are given by \(\displaystyle x= \frac{-5q\pm\sqrt{25q^2- 8q(5q- 3)}}{4q}\). The roots are real as long and distinct as the "discriminant" is strictly positive: \(\displaystyle 25q^2- 8q(5q- 3)= 24q- 15q^2= 3q(8- 5q) \ge 0\). The product of two numbers is positive if and only if the two numbers have the same sign- both positive or both negative.
Both are positive when 3q> 0 and 8- 5q> 0. From the second, 8>5q so q< 8/5. From the first, q> 0 so 0< q< 8/5.
Both are negative when 3q< 0 and 8- 5q< 0. From the second, 8< 5q so q> 8/5. From the first q< 0. It is impossible to satisfy both so this give no values of q.
0< q< 8/5.
