1. **State the problem:** Solve the quadratic equation $2x^2 - 2x = 112$ for $x$. 2. **Rewrite the equation:** Move all terms to one side to set the equation equal to zero: $$2x^2 - 2x - 112 = 0$$ 3. **Simplify the equation:** Divide every term by 2 to simplify: $$\cancel{2}x^2 - \cancel{2}x - \cancel{2}56 = 0 \implies x^2 - x - 56 = 0$$ 4. **Factor the quadratic:** Find two numbers that multiply to $-56$ and add to $-1$. These are $-8$ and $7$. $$x^2 - x - 56 = (x - 8)(x + 7) = 0$$ 5. **Solve for $x$:** Set each factor equal to zero: $$x - 8 = 0 \implies x = 8$$ $$x + 7 = 0 \implies x = -7$$ 6. **Final answer:** The solutions to the equation are $x = 8$ and $x = -7$.