1. **State the problem:** Factor the quadratic expression $$x^2 - 3x - 10$$ as the product of two binomials. 2. **Recall the factoring formula:** For a quadratic in the form $$x^2 + bx + c$$, we look for two numbers $$a$$ and $$b$$ such that: - Their product is $$c$$ - Their sum is $$b$$ 3. **Identify coefficients:** Here, $$b = -3$$ and $$c = -10$$. 4. **Find two numbers:** We need two numbers whose product is $$-10$$ and sum is $$-3$$. - Possible pairs for $$-10$$: $$(1, -10), (-1, 10), (2, -5), (-2, 5)$$ - Check sums: - $$1 + (-10) = -9$$ - $$-1 + 10 = 9$$ - $$2 + (-5) = -3$$ (this matches!) - $$-2 + 5 = 3$$ 5. **Write the factors:** Using $$2$$ and $$-5$$, the factorization is: $$x^2 - 3x - 10 = (x + 2)(x - 5)$$ 6. **Verify by expansion:** $$ (x + 2)(x - 5) = x^2 - 5x + 2x - 10 = x^2 - 3x - 10 $$ **Final answer:** $$x^2 - 3x - 10 = (x + 2)(x - 5)$$