Find the quotient and remainder using long division for
\[
\frac{x^{2}+4 x+6}{x+3}
\]
The quotient is

Step 1 ：Given the problem to find the quotient and remainder of the polynomial division \(\frac{x^{2}+4 x+6}{x+3}\).

Step 2 ：First, divide the leading term of the numerator by the leading term of the denominator. In this case, that means dividing \(x^2\) by \(x\), which gives \(x\). This is the first term of the quotient.

Step 3 ：Next, multiply the entire denominator by the term we just found (\(x\)) and subtract this from the original numerator. This gives us a new polynomial which we then divide by the denominator, repeating the process until we can't divide anymore.

Step 4 ：The final quotient is the sum of all the terms we found, and the remainder is the last polynomial we couldn't divide.

Step 5 ：\(\boxed{\text{Final Answer: The quotient is } x - 2 \text{ and the remainder is } 0}\). Therefore, the division of the polynomials is exact and can be written as \(x^{2}+4 x+6 = (x+3)(x - 2)\).