For a quadratic equation set equal to zero as in ax2+bx+c = 0, the discriminant is defined as b2-4ac. When the value of the discriminant is zero, the equation has one real double root. If a, b, c are integers (a≠0), then the double root is rational.
Writing your equation using the letter k (instead of a):
(k+4)x2-(3k-1)x+25 = 0, it is then the case that a = k+4, b=-3k+1, c = 25.
An expression for the value of the discriminant is then:
Simplifying and setting equal to 0:
9k2-6k+1-100k-400 = 0
That is another quadratic equation which can be solved by factoring or using the quadratic formula.
9x2+27x-133x-399=0 (split the middle)
9x(x+3)-133(x+3)=0 (factor by grouping)
x= 133/9 or x = -3
Those are the values of k (or a in the original problem) that will result in a solution that has exactly one double root.
For example when k=-3
x=-5 (multiplicity 2)
Left to you to try k = 133/9.