Divide. \[ \left(3 x^{2}+x^{3}+4+5 x\right) \div(x+1) \]

To divide \( (3x^2 + x^3 + 4 + 5x) \) by \( (x + 1) \), we can first rearrange the polynomial in standard form: \[ x^3 + 3x^2 + 5x + 4 \] Now we can use polynomial long division: 1. Divide the leading term of the dividend \( x^3 \) by the leading term of the divisor \( x \) to get \( x^2 \). 2. Multiply \( x^2 \) by \( (x + 1) \) to get \( x^3 + x^2 \). 3. Subtract \( (x^3 + x^2) \) from \( (x^3 + 3x^2 + 5x + 4) \): \[ (x^3 + 3x^2 + 5x + 4) - (x^3 + x^2) = (3x^2 - x^2) + (5x) + (4) = 2x^2 + 5x + 4 \] 4. Now, bring down the next term \( 2x^2 + 5x + 4 \). Divide \( 2x^2 \) by \( x \) to get \( 2x \). 5. Multiply \( 2x \) by \( (x + 1) \) to get \( 2x^2 + 2x \). 6. Subtract \( (2x^2 + 2x) \) from \( (2x^2 + 5x + 4) \): \[ (2x^2 + 5x + 4) - (2x^2 + 2x) = (5x - 2x) + 4 = 3x + 4 \] 7. Now bring down \( 3x + 4 \). Divide \( 3x \) by \( x \) to get \( 3 \). 8. Multiply \( 3 \) by \( (x + 1) \) to get \( 3x + 3 \). 9. Subtract \( (3x + 3) \) from \( (3x + 4) \): \[ (3x + 4) - (3x + 3) = 4 - 3 = 1 \] So we have reached a remainder of \( 1 \). Putting it all together, the result of the division can be written as: \[ x^2 + 2x + 3 + \frac{1}{x + 1} \] Thus, the final answer to the division is: \[ \boxed{x^2 + 2x + 3 + \frac{1}{x + 1}} \]