Recall that for a typical quadratic equation ax² + bx + c = 0,

Sum of roots = -b/a
Product of roots = c/a
Discriminant = b² - 4ac

x = ( -b±√(b²-4ac) ) / 2a


Now it is given that 2ax² + (2a+b)x + b = 0

So discriminant (b² - 4ac)

= (2a+b)² - 4(2a)(b)
= (2a)² + 2(2a)(b) + b² - 4(2a)(b)
= (2a)² - 2(2a)b + b²
= (2a-b)²

(Recall that :
(x + y)² = x² + 2xy + y²
(x - y)² = x² - 2xy + y²

So (x - y)² = (x + y)² - 4xy and conversely (x + y)² = (x - y)² + 4xy )


Since (2a-b)² ≥ 0 for all real values of a and b, (recall that the square of any real value is always positive or 0)

the equation has roots for all values of a and b.


If the sum of roots is -5, then :

-(2a+b) / 2a = -5
(2a+b) / 2a = 5
2a + b = 5 × 2a
2a + b = 10a
b = 10a - 2a = 8a