1. **State the problem:** Solve the quadratic equation given as $\left(x:2\right) (2x+3) = 0$. 2. **Rewrite the equation:** The notation $x:2$ is interpreted as $\frac{x}{2}$. So the equation becomes: $$\frac{x}{2} (2x + 3) = 0$$ 3. **Use the zero product property:** If a product of two factors is zero, then at least one of the factors must be zero. So: $$\frac{x}{2} = 0 \quad \text{or} \quad 2x + 3 = 0$$ 4. **Solve each factor:** - For $\frac{x}{2} = 0$: $$x = 0$$ - For $2x + 3 = 0$: $$2x = -3$$ $$x = -\frac{3}{2}$$ 5. **Final answer:** The solutions to the quadratic equation are: $$x = 0 \quad \text{or} \quad x = -\frac{3}{2}$$