1. **State the problem:** Simplify or factor the quadratic expression $2x^2 - 3x - 35$. 2. **Recall the factoring formula:** For a quadratic $ax^2 + bx + c$, we look for two numbers that multiply to $a \times c$ and add to $b$. 3. Here, $a=2$, $b=-3$, and $c=-35$. Calculate $a \times c = 2 \times (-35) = -70$. 4. Find two numbers that multiply to $-70$ and add to $-3$. These numbers are $7$ and $-10$ because $7 \times (-10) = -70$ and $7 + (-10) = -3$. 5. Rewrite the middle term using these numbers: $$2x^2 + 7x - 10x - 35$$ 6. Group terms: $$ (2x^2 + 7x) - (10x + 35) $$ 7. Factor each group: $$ x(2x + 7) - 5(2x + 7) $$ 8. Factor out the common binomial: $$ (2x + 7)(x - 5) $$ **Final answer:** The factorization of $2x^2 - 3x - 35$ is $$ (2x + 7)(x - 5) $$.