Let's break down and simplify the expression:

$X + \frac{7}{7x + 35} \times \frac{x^2 - 3x - 40}{x - 8}$

1. Factor the denominators and numerators:

   • The first fraction involves $7x + 35$. We can factor out a 7: $7x + 35 = 7(x + 5)$ So, the fraction becomes: $\frac{7}{7x + 35} = \frac{7}{7(x + 5)} = \frac{1}{x + 5}$

2. Factor the quadratic expression $x^2 - 3x - 40$:

   • To factor $x^2 - 3x - 40$, we need two numbers that multiply to -40 and add to -3. These numbers are -8 and 5. So we factor the quadratic as: $x^2 - 3x - 40 = (x - 8)(x + 5)$

3. Rewrite the expression using these factored forms: $X + \frac{1}{x + 5} \times \frac{(x - 8)(x + 5)}{x - 8}$

4. Simplify the expression:

   • The $(x - 8)$ terms cancel out: $X + \frac{(x + 5)}{x + 5}$
   • The $(x + 5)$ terms also cancel out, leaving: $X + 1$

Thus, the simplified expression is: $X + 1$

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Would you like more details on any of the steps, or do you have another question? Here are a few questions to dive deeper:

1. What is the significance of factoring polynomials in simplifying rational expressions?
2. How do you recognize when to cancel terms in fractions?
3. How does simplifying rational expressions help in solving equations?
4. Can you apply this same technique to more complex rational expressions?
5. What role does factoring play in finding the domain of a rational expression?

Tip: When simplifying rational expressions, always look for common factors between the numerator and denominator to cancel out!