Sum of roots is -b and product of roots is -4, so if one root is -4, the other must be 1 and -b=-4+1=-3, so b=3. The quadratic equation is x²+3x-4=(x+4)(x-1)=0.
If x²+Px+Q=0 is a perfect square (equal roots) then (P/2)²=Q, and P=±2√Q, so there would be many solutions for P and Q. Q must be a perfect square: 1, 4, 9, 16, ...
And the corresponding values for P are ±2, ±4, ±6, ±8, ...
If b in the first equation is the same as P, then P=3 and Q=9/4.
If Q is the square of the roots of the first equation, then Q is 16 or 1 and P=8 or -2.