1. **State the problem:** Solve the equation $$(z+1)(z+3)=0$$ for $z$. 2. **Formula and rule:** The zero product property states that if the product of two factors is zero, then at least one of the factors must be zero. That is, if $$ab=0$$, then $$a=0$$ or $$b=0$$. 3. **Apply the zero product property:** Set each factor equal to zero: $$z+1=0$$ $$z+3=0$$ 4. **Solve each equation:** For $$z+1=0$$: $$z=-1$$ For $$z+3=0$$: $$z=-3$$ 5. **Final answer:** The solutions to the equation $$(z+1)(z+3)=0$$ are $$z=-1$$ and $$z=-3$$. These correspond to the points where the product equals zero, matching the roots of the equation and the x-intercepts of the related graph.