1. **State the problem:** We are given that $7y$ is directly proportional to $(x+3)^2$, and the sum of values of $y$ when $x=8$ and $x=10$ is 870. We need to find an equation connecting $x$ and $y$. 2. **Express the direct proportionality:** Since $7y$ is directly proportional to $(x+3)^2$, we write: $$7y = k(x+3)^2$$ where $k$ is the constant of proportionality. 3. **Rewrite the equation for $y$:** $$y = \frac{k}{7}(x+3)^2$$ 4. **Use the given sum condition:** When $x=8$, $$y = \frac{k}{7}(8+3)^2 = \frac{k}{7} \times 11^2 = \frac{121k}{7}$$ When $x=10$, $$y = \frac{k}{7}(10+3)^2 = \frac{k}{7} \times 13^2 = \frac{169k}{7}$$ 5. **Sum of $y$ values:** $$\frac{121k}{7} + \frac{169k}{7} = 870$$ $$\frac{290k}{7} = 870$$ 6. **Solve for $k$:** $$k = \frac{870 \times 7}{290} = \frac{6090}{290} = 21$$ 7. **Write the final equation connecting $x$ and $y$:** $$7y = 21(x+3)^2$$ or equivalently, $$y = 3(x+3)^2$$ **Final answer:** $$y = 3(x+3)^2$$