1. **State the problem:** Simplify the expression $$\frac{2x}{x^2 - 3x - 88} - \frac{2x - 1}{x^2 - 10x - 11}$$. 2. **Factor the denominators:** - Factor $$x^2 - 3x - 88$$. We look for two numbers that multiply to $$-88$$ and add to $$-3$$. These are $$-11$$ and $$8$$. $$x^2 - 3x - 88 = (x - 11)(x + 8)$$. - Factor $$x^2 - 10x - 11$$. We look for two numbers that multiply to $$-11$$ and add to $$-10$$. These are $$-11$$ and $$1$$. $$x^2 - 10x - 11 = (x - 11)(x + 1)$$. 3. **Rewrite the expression with factored denominators:** $$\frac{2x}{(x - 11)(x + 8)} - \frac{2x - 1}{(x - 11)(x + 1)}$$ 4. **Find the common denominator:** The common denominator is $$ (x - 11)(x + 8)(x + 1) $$. 5. **Rewrite each fraction with the common denominator:** $$\frac{2x(x + 1)}{(x - 11)(x + 8)(x + 1)} - \frac{(2x - 1)(x + 8)}{(x - 11)(x + 8)(x + 1)}$$ 6. **Combine the fractions:** $$\frac{2x(x + 1) - (2x - 1)(x + 8)}{(x - 11)(x + 8)(x + 1)}$$ 7. **Expand the numerators:** - $$2x(x + 1) = 2x^2 + 2x$$ - $$(2x - 1)(x + 8) = 2x^2 + 16x - x - 8 = 2x^2 + 15x - 8$$ 8. **Subtract the numerators:** $$2x^2 + 2x - (2x^2 + 15x - 8) = 2x^2 + 2x - 2x^2 - 15x + 8 = -13x + 8$$ 9. **Final simplified expression:** $$\frac{-13x + 8}{(x - 11)(x + 8)(x + 1)}$$ 10. **Optional: Factor numerator if possible:** The numerator $$-13x + 8$$ cannot be factored further simply. **Answer:** $$\boxed{\frac{-13x + 8}{(x - 11)(x + 8)(x + 1)}}$$