1. Stating the problem: Factorize the expression $$E = x^2 - 2x - 15$$. 2. Use the quadratic expression factorization formula: For $$ax^2 + bx + c$$, find two numbers that multiply to $$ac$$ and add to $$b$$. 3. Here, $$a=1$$, $$b=-2$$, and $$c=-15$$. We need two numbers that multiply to $$1 \times (-15) = -15$$ and add to $$-2$$. 4. The numbers are $$3$$ and $$-5$$ because $$3 \times (-5) = -15$$ and $$3 + (-5) = -2$$. 5. Rewrite the middle term using these numbers: $$x^2 + 3x - 5x - 15$$ 6. Group terms: $$(x^2 + 3x) - (5x + 15)$$ 7. Factor each group: $$x(x + 3) - 5(x + 3)$$ 8. Factor out the common binomial: $$(x - 5)(x + 3)$$ 9. Final answer: $$E = (x - 5)(x + 3)$$.