8. Write \( \frac{3 x^{2}+6 x}{2 x y+4 y} \times \frac{2 y}{9 x^{2}} \) in its simplified form and substitute the value to evaluate, given that \( x=1 \) and \( y=2 \).

First, let's simplify the expression \( \frac{3 x^{2}+6 x}{2 x y+4 y} \times \frac{2 y}{9 x^{2}} \). Start with the first part, \( \frac{3 x^{2}+6 x}{2 x y+4 y} \): 1. Factor the numerator: \( 3x(x+2) \). 2. Factor the denominator: \( 2y(x+2) \). Now, we can rewrite the first fraction: \[ \frac{3x(x+2)}{2y(x+2)} \] The \( (x+2) \) cancels out (as long as \( x \neq -2 \)), giving us: \[ \frac{3x}{2y} \] Now multiplying this by the second part \( \frac{2y}{9x^{2}} \): \[ \frac{3x}{2y} \times \frac{2y}{9x^{2}} = \frac{3x \cdot 2y}{2y \cdot 9x^{2}} = \frac{3 \cdot 2}{9x} = \frac{6}{9x} = \frac{2}{3x} \] Now substitute \( x=1 \) into \( \frac{2}{3x} \): \[ \frac{2}{3 \cdot 1} = \frac{2}{3} \] The simplified form of the expression is \( \frac{2}{3x} \) and the evaluated result when \( x=1 \) is \( \frac{2}{3} \).