Use synthetic division to perform the division. \[ \frac{x^{4}+x^{3}+4 x^{2}+5 x+1}{x+1} \]

Step 1 ：The given problem is to perform synthetic division on the polynomial \(x^{4} + x^{3} + 4x^{2} + 5x + 1\) by the divisor \(x + 1\).

Step 2 ：Synthetic division is a shorthand method of dividing polynomials where we divide the coefficients of the polynomial with the divisor.

Step 3 ：Performing the synthetic division, we get the quotient as \(x^{3} + 0x^{2} + 3x + 2\) and the remainder as \(-1\).

Step 4 ：\(\boxed{x^{3} + 3x + 2 - \frac{1}{x+1}}\) is the final result of the synthetic division.