Answer
$$x=3, \quad x=-1$$
Work Step by Step
We solve the equation by cross multiplying and factoring:
$$\frac{x+2}{4x+3}=\frac{1}{x} \quad \Rightarrow \quad x(x+2)=4x+3, \quad (x \neq - \frac{3}{4}, 0) \quad \Rightarrow \quad x^2-2x-3=0 \quad (x \neq - \frac{3}{4}, 0) \quad \Rightarrow \quad (x-3)(x+1)=0 \quad \Rightarrow \quad x=3, \quad x=-1$$
