\[ \frac{x^{3}+3 x^{2}+4 x+2}{x+1}=\square \] (Simplify your answer.)

Step 1 ：The given expression is \(\frac{x^{3}+3 x^{2}+4 x+2}{x+1}\).

Step 2 ：We are asked to simplify this expression.

Step 3 ：We can do this by performing polynomial division. The numerator is a cubic polynomial and the denominator is a linear polynomial.

Step 4 ：Using synthetic division, we find that the simplified expression is \(x^{2} + 2x + 2\) and the remainder is 0.

Step 5 ：This means that the original expression simplifies to \(x^{2} + 2x + 2\).

Step 6 ：Final Answer: \(\boxed{x^{2} + 2x + 2}\)