To simplify the expression $\frac{x^2 + 6x - 16}{2 - x}$, follow these steps:

1. Factor the numerator: We need to factor the quadratic expression $x^2 + 6x - 16$.

   • Look for two numbers that multiply to $-16$ and add to $6$. The numbers $8$ and $-2$ work because $8 \times (-2) = -16$ and $8 + (-2) = 6$.
   • So, $x^2 + 6x - 16$ factors as $(x + 8)(x - 2)$.

   Therefore: $\frac{x^2 + 6x - 16}{2 - x} = \frac{(x + 8)(x - 2)}{2 - x}$

2. Simplify the expression: Notice that $x - 2$ and $2 - x$ are related. Specifically, $2 - x$ is $-(x - 2)$. $\frac{(x + 8)(x - 2)}{-(x - 2)} = -\frac{(x + 8)(x - 2)}{x - 2}$

3. Cancel out the common factor: The $x - 2$ in the numerator and denominator cancels out, leaving: $-(x + 8)$

4. Final answer: $-(x + 8) = -x - 8$

So, $\frac{x^2 + 6x - 16}{2 - x} = -x - 8$, assuming $x \neq 2$ (since $x = 2$ would make the denominator zero, which is undefined).

Would you like to explore more details, or do you have any questions?

Here are some related questions:

1. How does factoring quadratics help in simplifying rational expressions?
2. What happens if the denominator is a more complex expression?
3. How do you determine when two expressions are equivalent?
4. Can you graph $-x - 8$ to see its behavior?
5. What are the consequences if $x = 2$ in this expression?

Tip: Always check for any restrictions on the variable after simplifying an expression, especially when canceling factors.