1. **State the problem:** Simplify the complex rational expression \n $$\frac{-\frac{7}{x+7} - \frac{2}{x+6}}{-\frac{6}{x+7}}$$\n and express the product \n $$(2 \times 10^{7})(4 \times 10^{3})$$\n in scientific notation.\n \n2. **Simplify the complex rational expression:**\n Start with the numerator: $$-\frac{7}{x+7} - \frac{2}{x+6}$$\n Find a common denominator for the numerator: $$(x+7)(x+6)$$\n Rewrite numerator terms: $$-\frac{7(x+6)}{(x+7)(x+6)} - \frac{2(x+7)}{(x+6)(x+7)} = \frac{-7(x+6) - 2(x+7)}{(x+7)(x+6)}$$\n Simplify numerator: $$-7x - 42 - 2x - 14 = -9x - 56$$\n So numerator is $$\frac{-9x - 56}{(x+7)(x+6)}$$\n Denominator is $$-\frac{6}{x+7}$$\n Rewrite the entire expression: $$\frac{\frac{-9x - 56}{(x+7)(x+6)}}{-\frac{6}{x+7}}$$\n 3. **Divide by a fraction by multiplying by its reciprocal:**\n $$= \frac{-9x - 56}{(x+7)(x+6)} \times \frac{x+7}{-6}$$\n 4. **Cancel common factors:**\n $$= \frac{-9x - 56}{\cancel{(x+7)}(x+6)} \times \frac{\cancel{x+7}}{-6} = \frac{-9x - 56}{x+6} \times \frac{1}{-6}$$\n 5. **Multiply numerators and denominators:**\n $$= \frac{-9x - 56}{x+6} \times \frac{1}{-6} = \frac{(-9x - 56) \times 1}{(x+6) \times (-6)} = \frac{-9x - 56}{-6(x+6)}$$\n 6. **Simplify signs:**\n $$= \frac{-9x - 56}{-6(x+6)} = \frac{9x + 56}{6(x+6)}$$\n This is the simplified form of the complex rational expression.\n \n7. **Simplify the product in scientific notation:**\n $$(2 \times 10^{7})(4 \times 10^{3}) = (2 \times 4) \times 10^{7+3} = 8 \times 10^{10}$$\n \n**Final answers:**\n - Simplified expression: $$\frac{9x + 56}{6(x+6)}$$\n- Product in scientific notation: $$8 \times 10^{10}$$