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 \( y=2 \)

To simplify the expression \( \frac{3 x^{2}+6 x}{2 x y+4 y} \times \frac{2 y}{9 x^{2}} \): 1. Factor the numerator and denominator of the first fraction: - The numerator \( 3x^2 + 6x = 3x(x + 2) \). - The denominator \( 2xy + 4y = 2y(x + 2) \). So, the fraction becomes \( \frac{3x(x + 2)}{2y(x + 2)} \). 2. Simplifying the above fraction, the \( (x + 2) \) terms cancel out (as long as \( x + 2 \neq 0 \)): \( \frac{3x}{2y} \). 3. Next, we incorporate the second fraction: - Thus, we have \( \frac{3x}{2y} \times \frac{2y}{9x^2} \). 4. Multiplying these fractions yields: \( \frac{3x \cdot 2y}{2y \cdot 9x^2} = \frac{6x}{18x^2} = \frac{1}{3x} \) (after canceling \( 2y \) and simplifying). Now, substituting \( x = 1 \): \( \frac{1}{3(1)} = \frac{1}{3} \). Therefore, the simplified expression evaluates to \( \frac{1}{3} \).