jmc

We are given that $\frac{x+2}{2x+2}=\frac{4x+3}{7x+3}$. Cross multiplying, we find $(x+2)(7x+3)=(2x+2)(4x+3)$. Multiplying out each side by the distributive property yields $7x^2+3x+14x+6=8x^2+6x+8x+6$. Simplifying, we find $x^2=3x$, so $x=0,3$. Checking our two answers, we find that indeed $\frac{2}{2}=\frac{3}{3}$, and also that $\frac{5}{8}=\frac{15}{24}$. The product of our two solutions is $0\cdot 3=\boxed{0}$.