Answer
$f+g =3x^2+4x+1$, domain $\{x|x=all\ real\ numbers \}$
$f-g= 3x^2-2x+1$, domain $\{x|x=all\ real\ numbers \}$
$f\cdot g= 9x^3+3x^2+3x$, domain $\{x|x=all\ real\ numbers \}$
$\frac{f}{g}=\frac{3x^2+x+1}{3x}$, domain $\{x|x\ne0 \}$
Work Step by Step
Step 1. Given $f(x)=3x^2+x+1, g(x)=3x$, we have $f+g=(3x^2+x+1)+(3x)=3x^2+4x+1$, domain $\{x|x=all\ real\ numbers \}$
Step 2. We have $f-g=(3x^2+x+1)-(3x)=3x^2-2x+1$, domain $\{x|x=all\ real\ numbers \}$
Step 3. We have $f\cdot g=(3x^2+x+1)(3x)=9x^3+3x^2+3x$, domain $\{x|x=all\ real\ numbers \}$
Step 4. We have $\frac{f}{g}=\frac{3x^2+x+1}{3x}$, domain $\{x|x\ne0 \}$
