1. **State the problem:** Evaluate the expression $$E = \frac{-4(3) + 7(-2) + 20}{-5 + 2(-8) + (-3)(4)}$$. 2. **Apply multiplication:** Calculate each product inside the numerator and denominator. Numerator: $$-4(3) = -12$$, $$7(-2) = -14$$. Denominator: $$2(-8) = -16$$, $$(-3)(4) = -12$$. 3. **Substitute these values back:** $$E = \frac{-12 - 14 + 20}{-5 - 16 - 12}$$ 4. **Simplify numerator and denominator separately:** Numerator: $$-12 - 14 + 20 = (-12 - 14) + 20 = -26 + 20 = -6$$ Denominator: $$-5 - 16 - 12 = (-5 - 16) - 12 = -21 - 12 = -33$$ 5. **Rewrite the fraction:** $$E = \frac{-6}{-33}$$ 6. **Simplify the fraction by dividing numerator and denominator by their greatest common divisor (3):** $$E = \frac{\cancel{-6}^{\div 3}}{\cancel{-33}^{\div 3}} = \frac{-2}{-11}$$ 7. **Simplify signs:** Negative divided by negative is positive, so $$E = \frac{2}{11}$$ **Final answer:** $$E = \frac{2}{11}$$