1. **Problem Statement:** Find the value of $x$ given two polygons with sides labeled $x+5$, 4 and 5, 15 respectively, assuming the polygons are similar and corresponding sides are proportional. 2. **Formula and Rules:** For similar polygons, corresponding sides are proportional. This means: $$\frac{x+5}{5} = \frac{4}{15}$$ 3. **Set up the proportion:** $$\frac{x+5}{5} = \frac{4}{15}$$ 4. **Cross multiply:** $$15(x+5) = 5 \times 4$$ 5. **Simplify both sides:** $$15x + 75 = 20$$ 6. **Isolate $x$:** $$15x = 20 - 75$$ $$15x = -55$$ 7. **Divide both sides by 15:** $$x = \frac{-55}{15}$$ 8. **Simplify the fraction:** $$x = \frac{\cancel{-55}^{\times 11}}{\cancel{15}^{\times 3}} = -\frac{11}{3}$$ **Final answer:** $$x = -\frac{11}{3}$$