1. **State the problem:** Calculate the value of the expression $$E = \frac{-4(3) + 7(-2) + 20}{-5 + 2(-8) + (-3)(4)}$$. 2. **Apply multiplication in numerator and denominator:** $$\text{Numerator} = -4 \times 3 + 7 \times (-2) + 20 = -12 - 14 + 20$$ $$\text{Denominator} = -5 + 2 \times (-8) + (-3) \times 4 = -5 - 16 - 12$$ 3. **Simplify numerator and denominator:** $$\text{Numerator} = -12 - 14 + 20 = (-12 - 14) + 20 = -26 + 20 = -6$$ $$\text{Denominator} = -5 - 16 - 12 = (-5 - 16) - 12 = -21 - 12 = -33$$ 4. **Write the fraction with simplified numerator and denominator:** $$E = \frac{-6}{-33}$$ 5. **Simplify the fraction by dividing numerator and denominator by their greatest common divisor (3):** $$E = \frac{\cancel{-6}^{\div 3}}{\cancel{-33}^{\div 3}} = \frac{-2}{-11}$$ 6. **Simplify signs:** $$E = \frac{-2}{-11} = \frac{2}{11}$$ **Final answer:** $$E = \frac{2}{11}$$