$a)$

$f\left(x\right)=x^2+bx+c$  

$x_1=-1$  

$x_2=4$  

$f\left(x\right)=\left(x+1\right)\left(x-4\right)=x^2-3x-4$  

$b=-3$  

$c=-4$  

 

$b)$  

$f\left(x\right)=\left(x-\frac{1}{2}\right)\left(x-\frac{1}{3}\right)=x^2-\frac{5}{6}x+\frac{1}{6}$  

$b=-\frac{5}{6}$  

$c=\frac{1}{6}$  

 

$c)$  

$f\left(x\right)=\left(x-\sqrt{2}\right)\left(x+2\sqrt{2}\right)=x^2+2\sqrt{2}x-\sqrt{2}x-\sqrt{2}\cdot 2\sqrt{2}=x^2+\sqrt{2}x-4$  

$b=\sqrt{2}$  

$c=-4$