The roots of a quadratic equation ax^2 + bx + c = 0 are real if `b^2 - 4ac >= 0`
The roots of the equation ax^2 + a^2x + a^3 = 0 are real if `(a^2)^2 - 4*a*a^3 >= 0`
=> `a^4 - 4a^4 >= 0`
=> `a^4 >= 4a^4`
a^4 is a positive number and so is 4. This makes `4a^4 > a^4` for all values of a.
For no value of a does the given equation have real roots.