1. **State the problem:** We need to find which value of $x$ satisfies the equation $$(x - 3)(x - 5) = 35.$$\n\n2. **Use the formula:** The equation is a product of two binomials equal to 35. We can expand and simplify it or test each option.\n\n3. **Expand the left side:**\n$$ (x - 3)(x - 5) = x^2 - 5x - 3x + 15 = x^2 - 8x + 15. $$\n\n4. **Rewrite the equation:**\n$$ x^2 - 8x + 15 = 35. $$\n\n5. **Bring all terms to one side:**\n$$ x^2 - 8x + 15 - 35 = 0 \implies x^2 - 8x - 20 = 0. $$\n\n6. **Solve the quadratic equation:**\nUse the quadratic formula $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ where $a=1$, $b=-8$, and $c=-20$.\n\n7. **Calculate the discriminant:**\n$$ b^2 - 4ac = (-8)^2 - 4(1)(-20) = 64 + 80 = 144. $$\n\n8. **Find the roots:**\n$$ x = \frac{-(-8) \pm \sqrt{144}}{2(1)} = \frac{8 \pm 12}{2}. $$\n\n9. **Calculate each root:**\n- $$ x = \frac{8 + 12}{2} = \frac{20}{2} = 10. $$\n- $$ x = \frac{8 - 12}{2} = \frac{-4}{2} = -2. $$\n\n10. **Check which options match the roots:**\nGiven options are $-8$, $-5$, $2$, and $10$. Only $10$ is a root of the equation.\n\n**Final answer:** $x = 10$ satisfies the equation.