1. **State the problem:** Solve the equation $ (x - 3)(x - 5) = 35 $ for $ x $.\n\n2. **Use the distributive property (FOIL) to expand the left side:**\n$$ (x - 3)(x - 5) = x^2 - 5x - 3x + 15 = x^2 - 8x + 15 $$\n\n3. **Rewrite the equation:**\n$$ x^2 - 8x + 15 = 35 $$\n\n4. **Bring all terms to one side to set the equation to zero:**\n$$ x^2 - 8x + 15 - 35 = 0 $$\n$$ x^2 - 8x - 20 = 0 $$\n\n5. **Solve the quadratic equation $ x^2 - 8x - 20 = 0 $ using the quadratic formula:**\nThe quadratic formula is\n$$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$\nwhere $ a=1 $, $ b=-8 $, and $ c=-20 $.\n\n6. **Calculate the discriminant:**\n$$ b^2 - 4ac = (-8)^2 - 4(1)(-20) = 64 + 80 = 144 $$\n\n7. **Calculate the roots:**\n$$ x = \frac{-(-8) \pm \sqrt{144}}{2(1)} = \frac{8 \pm 12}{2} $$\n\n8. **Find the two solutions:**\n- For the plus sign:\n$$ x = \frac{8 + 12}{2} = \frac{20}{2} = 10 $$\n- For the minus sign:\n$$ x = \frac{8 - 12}{2} = \frac{-4}{2} = -2 $$\n\n9. **Check which of the given options are solutions:**\n- $ x = 10 $ is a solution.\n- $ x = -8, -5, 2 $ are not solutions.\n\n**Final answer:** $ x = 10 $ is a solution to the equation.