Step 1 :Given the equation of the circle as $x^{2}+y^{2}+6 x+4 y-3=0$
Step 2 :Rearrange the equation as $x^{2}+6 x+y^{2}+4 y-3=0$
Step 3 :Group the x terms and y terms together to get $(x^{2}+6 x)+(y^{2}+4 y)=3$
Step 4 :Complete the square for the x terms and y terms to get $(x^{2}+6 x+9)+(y^{2}+4 y+4)=3+9+4$
Step 5 :Simplify the equation to get $(x+3)^{2}+(y+2)^{2}=4^{2}$
Step 6 :From the equation, we can see that the center of the circle is at the point (-3, -2)
Step 7 :\(\boxed{\text{The correct completion of Mrs. Culland's work is }"(x+3)^{2}+(y+2)^{2}=4^{2}", \text{ so the center is } (-3,-2)}\)