The equation $3x^2+4ax+b=0$  has atleast one root in [0,1]if :

Consider the function $f\left(x\right)=x^3+2a{x}^2+bx$

Obviously $f\left(x\right)$  is continuous in [0,1] and differentiable in ]0,1[.

Also $f\left(0\right)=0$  if $f\left(1\right)=1+2a+b=0$ , then all conditions of Rolle’s theorem are satisfied

$\therefore f\text{'}\left(c\right)=0$  for atleast one $c$ in ]0,1[

Hence, $f\text{'}\left(x\right)=3x^2+4ax+b=0$  at least once in ]0,1[

i.e., the equation $3x^2+4ax+b=0$  has atleast one root in ]0,1[.

So $2a+b+1=0$ .