Simplify each expression. \( 3+\frac{x+3}{6 x^{2}-18 x} \)

To simplify the expression \( 3+\frac{x+3}{6 x^{2}-18 x} \), we first factor the denominator. Notice that we can factor out \( 6x \): \[ 6x^2 - 18x = 6x(x - 3) \] Now, we can rewrite the original expression: \[ 3 + \frac{x + 3}{6x(x - 3)} \] To combine these two terms, we need a common denominator. The common denominator is \( 6x(x - 3) \). Rewrite \( 3 \) with this common denominator: \[ 3 = \frac{3 \cdot 6x(x - 3)}{6x(x - 3)} = \frac{18x(x - 3)}{6x(x - 3)} \] Now we can combine the two fractions: \[ \frac{18x(x - 3) + (x + 3)}{6x(x - 3)} \] Simplifying the numerator: \[ 18x(x - 3) + (x + 3) = 18x^2 - 54x + x + 3 = 18x^2 - 53x + 3 \] Thus, we have: \[ \frac{18x^2 - 53x + 3}{6x(x - 3)} \] The expression is now simplified to: \[ 3 + \frac{x + 3}{6x^2 - 18x} = \frac{18x^2 - 53x + 3}{6x(x - 3)} \]