Divide the polynomials. \[ \frac{3 x^{2}+5 x-2}{x+6}=\square \]

To divide the polynomials \( \frac{3x^{2} + 5x - 2}{x + 6} \), we can use polynomial long division. First, we divide the leading term \(3x^2\) by \(x\) from the divisor \(x + 6\) to get \(3x\). Next, we multiply \(3x\) by \(x + 6\), resulting in \(3x^2 + 18x\). We subtract this from the original polynomial: \[ (3x^2 + 5x - 2) - (3x^2 + 18x) = (5x - 18x - 2) = -13x - 2. \] Now, we repeat the process. We divide \(-13x\) by \(x\) to get \(-13\). We multiply \(-13\) by \(x + 6\) to get \(-13x - 78\). Now, we subtract this from our current remainder: \[ (-13x - 2) - (-13x - 78) = (-2 + 78) = 76. \] Therefore, the division of the polynomials can be summarized as: \[ 3x - 13 + \frac{76}{x + 6}. \] So the final answer is: \[ \frac{3x^2 + 5x - 2}{x + 6} = 3x - 13 + \frac{76}{x + 6}. \]