Simplify the following expression: \[ \frac{3 x+x^{2}}{18+6 x} \]

To simplify the expression \[ \frac{3 x + x^{2}}{18 + 6 x}, \] we start by factoring both the numerator and the denominator. For the numerator \(3x + x^2\), we can factor out \(x\): \[ 3x + x^2 = x(3 + x). \] For the denominator \(18 + 6x\), we can factor out 6: \[ 18 + 6x = 6(3 + x). \] Now, we can rewrite the expression with these factored forms: \[ \frac{x(3 + x)}{6(3 + x)}. \] Notice that \(3 + x\) appears in both the numerator and the denominator, allowing us to cancel: \[ \frac{x}{6} \quad \text{for } 3 + x \neq 0. \] Thus, the simplified expression is \[ \frac{x}{6}. \]