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The problem asks to write the following algebraic expression as a single fraction:

\[
\frac{x^2 + 7x - 18}{2x^2 - x - 6}
\]

### Step 1: Factor the numerator and denominator.
#### Numerator: \(x^2 + 7x - 18\)
We need to factor this quadratic. We are looking for two numbers that multiply to \(-18\) and add up to \(7\).

The factors are \(9\) and \(-2\), so:
\[
x^2 + 7x - 18 = (x + 9)(x - 2)
\]

#### Denominator: \(2x^2 - x - 6\)
We factor this quadratic expression. We are looking for two numbers that multiply to \(2 \times -6 = -12\) and add to \(-1\) (the coefficient of \(x\)).

The factors are \(3\) and \(-4\). We split the middle term:
\[
2x^2 - x - 6 = 2x^2 - 4x + 3x - 6
\]
Now factor by grouping:
\[
= 2x(x - 2) + 3(x - 2)
\]
Factor out \((x - 2)\):
\[
= (x - 2)(2x + 3)
\]

### Step 2: Write the expression as a single fraction.
Now that both numerator and denominator are factored, substitute these factored forms back into the original expression:

\[
\frac{(x + 9)(x - 2)}{(x - 2)(2x + 3)}
\]

### Step 3: Simplify the fraction.
Cancel out the common factor \((x - 2)\) from both the numerator and denominator:

\[
= \frac{x + 9}{2x + 3}
\]

Thus, the simplified single fraction is:

\[
\boxed{\frac{x + 9}{2x + 3}}
\]

Would you like more details on any step or have any questions? Here are a few related questions to further explore this topic:

1. How do you factor quadratics in general?
2. What if the common factor in the numerator and denominator is a more complex expression?
3. Can you explain why it’s important to check for excluded values (like \(x = 2\) in this case)?
4. How do you apply the distributive property to check factored forms?
5. What other methods exist for simplifying complex rational expressions?

**Tip:** Always verify factored forms by expanding them back to the original expression to ensure accuracy.

Write the following algebraic expression as a single fraction: (x^2 + 7x - 18) / (2x^2 - x - 6)

Learn how to simplify the rational expression (x^2 + 7x - 18) / (2x^2 - x - 6) by factoring both the numerator and the denominator. This step-by-step guide covers key algebraic techniques such as factoring quadratics and simplifying fractions.

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