1. **State the problem:** Solve the equation $x(x + 1) - 56 = 4x(x - 7)$ and find the sum of its solutions. 2. **Write the equation:** $$x(x + 1) - 56 = 4x(x - 7)$$ 3. **Expand both sides:** $$x^2 + x - 56 = 4x^2 - 28x$$ 4. **Bring all terms to one side to set the equation to zero:** $$x^2 + x - 56 - 4x^2 + 28x = 0$$ 5. **Combine like terms:** $$-3x^2 + 29x - 56 = 0$$ 6. **Multiply the entire equation by -1 to simplify:** $$3x^2 - 29x + 56 = 0$$ 7. **Recall the quadratic formula:** For $ax^2 + bx + c = 0$, the sum of solutions is given by $-\frac{b}{a}$. 8. **Identify coefficients:** $a = 3$, $b = -29$, $c = 56$ 9. **Calculate the sum of solutions:** $$-\frac{b}{a} = -\frac{-29}{3} = \frac{29}{3}$$ **Final answer:** The sum of the solutions is $\boxed{\frac{29}{3}}$.