1. **State the problem:** We are given two similar triangles $\triangle AFP \sim \triangle HDG$ with side lengths $AP = 15$, $FP = 11$, $DG = -8x - 7$, and $GH = -6x + 4$. We need to write an equation relating the side lengths and solve for $x$. 2. **Write the proportion from similarity:** Since the triangles are similar, corresponding sides are proportional: $$\frac{AP}{GH} = \frac{FP}{DG}$$ Substitute the given lengths: $$\frac{15}{-6x + 4} = \frac{11}{-8x - 7}$$ 3. **Cross multiply to solve for $x$:** $$15(-8x - 7) = 11(-6x + 4)$$ 4. **Distribute:** $$-120x - 105 = -66x + 44$$ 5. **Bring all terms involving $x$ to one side and constants to the other:** $$-120x + 66x = 44 + 105$$ $$-54x = 149$$ 6. **Divide both sides by $-54$:** $$x = \frac{149}{\cancel{-54}} \cancel{-1} = -\frac{149}{54}$$ 7. **Calculate the decimal value:** $$x \approx -2.76$$ **Final answer:** $$\boxed{x \approx -2.76}$$