first, Delete these brackets on the right side:
[tex]6x^{2}+bx+6=(3x+m)(nx+3) \\ 6x^{2}+bx+6=3nx^{2}+9x+nmx+3m \\ 6x^{2}+bx+6=3nx^{2}+(9+nm)x+3m[/tex]
Left side and right side are equal, so each parameters have to be equal. Next to "x^2" you've got parameters "6" and "3n". So:
3n=6 /:3
n=3
Free terms are "6" and "3m", so:
3m=6 /:3
m=2
Next to "x" you've got "b" and "9+nm" . We know, that m=2 and n=3 so:
b= 9+nm
b=9+2*3=9+6
So b=15