We need to factor the given expression:

\[
\frac{2x^2 + 14x - 16}{x^2 - 9x + 8}
\]

### Step 1: Factor the numerator

The numerator is \(2x^2 + 14x - 16\). First, we can factor out the greatest common factor (GCF), which is 2:

\[
2(x^2 + 7x - 8)
\]

Now, factor \(x^2 + 7x - 8\) by finding two numbers that multiply to \(-8\) and add up to \(7\). These numbers are \(8\) and \(-1\), so:

\[
x^2 + 7x - 8 = (x + 8)(x - 1)
\]

Thus, the numerator becomes:

\[
2(x + 8)(x - 1)
\]

### Step 2: Factor the denominator

The denominator is \(x^2 - 9x + 8\). We need two numbers that multiply to \(8\) and add to \(-9\). These numbers are \(-8\) and \(-1\), so:

\[
x^2 - 9x + 8 = (x - 8)(x - 1)
\]

### Step 3: Simplify the expression

Now we have:

\[
\frac{2(x + 8)(x - 1)}{(x - 8)(x - 1)}
\]

Since \(x - 1\) appears in both the numerator and the denominator, we can cancel it out (as long as \(x \neq 1\)):

\[
\frac{2(x + 8)}{x - 8}
\]

### Final Answer:
\[
\frac{2(x + 8)}{x - 8}
\]

### Do you want details or have any questions?

Here are five related questions to expand your understanding:

1. How do you factor a quadratic expression in general?
2. What happens if \(x = 1\) in the original expression?
3. Can we factor out a negative sign from a polynomial? If so, how?
4. What is the process of simplifying rational expressions?
5. How can we check if our factorization is correct?

**Tip:** Always check for common factors and cancellations when simplifying rational expressions.

Learn how to factor and simplify the rational expression (2x^2+14x-16)/(x^2-9x+8). This problem involves key algebraic concepts such as factoring quadratics and simplifying rational expressions. Suitable for students in Grades 9-11.

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