To solve this expression:

$\left( \frac{6x + 7}{x + 6} \times \frac{x^2 + 4x - 21}{6x^2 - 11x - 21} \right) - \frac{2}{x + 2}$

we will break it down into parts. First, we'll simplify each individual fraction and perform the operations step by step.

Step 1: Factor the expressions where possible.

• The expression $x^2 + 4x - 21$ factors as:

$x^2 + 4x - 21 = (x + 7)(x - 3)$

• The expression $6x^2 - 11x - 21$ factors as:

$6x^2 - 11x - 21 = (2x + 3)(3x - 7)$

Now, the expression becomes:

$\left( \frac{6x + 7}{x + 6} \times \frac{(x + 7)(x - 3)}{(2x + 3)(3x - 7)} \right) - \frac{2}{x + 2}$

Step 2: Simplify the product of the two fractions.

Multiply the numerators and denominators:

$\frac{(6x + 7)(x + 7)(x - 3)}{(x + 6)(2x + 3)(3x - 7)}$

At this stage, we cannot simplify further without knowing specific values for $x$, so we leave it in this factored form.

Step 3: Combine the terms.

Now we subtract the second fraction:

$\frac{(6x + 7)(x + 7)(x - 3)}{(x + 6)(2x + 3)(3x - 7)} - \frac{2}{x + 2}$

To subtract, we need a common denominator. The least common denominator (LCD) between $(x + 6)(2x + 3)(3x - 7)$ and $x + 2$ is:

$(x + 6)(2x + 3)(3x - 7)(x + 2)$

Thus, rewrite each fraction with this common denominator:

$\frac{(6x + 7)(x + 7)(x - 3)}{(x + 6)(2x + 3)(3x - 7)} - \frac{2(x + 6)(2x + 3)(3x - 7)}{(x + 6)(2x + 3)(3x - 7)(x + 2)}$

Step 4: Final expression.

Now, the numerator becomes:

$(6x + 7)(x + 7)(x - 3) - 2(x + 6)(2x + 3)(3x - 7)$

And the denominator is:

$(x + 6)(2x + 3)(3x - 7)(x + 2)$

Conclusion:

This expression is the final simplified form. To complete your answer, you need to identify the values of $x$ that would make the denominator equal to zero. These values exclude from the domain of the expression.

Set each factor in the denominator equal to zero and solve for $x$:

1. $x + 6 = 0 \Rightarrow x = -6$
2. $2x + 3 = 0 \Rightarrow x = -\frac{3}{2}$
3. $3x - 7 = 0 \Rightarrow x = \frac{7}{3}$
4. $x + 2 = 0 \Rightarrow x = -2$

Thus, the values that must be excluded from the domain are:

$x \neq -6, -\frac{3}{2}, \frac{7}{3}, -2$

Final Answer:

$\left( \frac{6x + 7}{x + 6} \times \frac{x^2 + 4x - 21}{6x^2 - 11x - 21} \right) - \frac{2}{x + 2} = \frac{(6x + 7)(x + 7)(x - 3) - 2(x + 6)(2x + 3)(3x - 7)}{(x + 6)(2x + 3)(3x - 7)(x + 2)}, \text{ where } x \neq -6, -\frac{3}{2}, \frac{7}{3}, -2$

Would you like me to expand the numerator and simplify further, or is this level of detail sufficient for you? Let me know if you need anything clarified!

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Further Questions:

1. What happens if we substitute specific values of $x$ into the expression?
2. How would you approach simplifying rational expressions involving trigonometric functions?
3. How do you find the least common denominator (LCD) when dealing with more complex fractions?
4. Can you simplify rational expressions without factoring, and when would that be appropriate?
5. What are the steps to solving rational equations after simplifying the expression?

Tip:

Always factor polynomials completely when working with rational expressions to make simplifying and canceling terms easier.