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The given fraction is:

\[
\frac{2x^2 + 11x - 40}{x^2 + 5x - 24}
\]

We are asked to fully simplify this expression by factoring both the numerator and the denominator, and then canceling common factors, if any.

### Step 1: Factor the numerator
The numerator is \( 2x^2 + 11x - 40 \).

We need to find two numbers that multiply to \( 2 \times -40 = -80 \) and add up to \( 11 \). These numbers are \( 16 \) and \( -5 \). So, we can rewrite the middle term and factor by grouping:

\[
2x^2 + 16x - 5x - 40 = 2x(x + 8) - 5(x + 8) = (2x - 5)(x + 8)
\]

Thus, the numerator factors as \( (2x - 5)(x + 8) \).

### Step 2: Factor the denominator
The denominator is \( x^2 + 5x - 24 \).

We need two numbers that multiply to \( -24 \) and add up to \( 5 \). These numbers are \( 8 \) and \( -3 \), so we can factor the denominator as:

\[
x^2 + 5x - 24 = (x + 8)(x - 3)
\]

### Step 3: Simplify the fraction
Now that we have factored both the numerator and the denominator, the expression becomes:

\[
\frac{(2x - 5)(x + 8)}{(x + 8)(x - 3)}
\]

We can cancel the common factor of \( (x + 8) \), leaving:

\[
\frac{2x - 5}{x - 3}
\]

### Final Answer:
The fully simplified expression is:

\[
\frac{2x - 5}{x - 3}
\]

Would you like a more detailed explanation or have any other questions?

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Here are 5 related questions to further explore the topic:

1. How do you factor a quadratic expression when the coefficient of \( x^2 \) is greater than 1?
2. What happens if the quadratic expressions in the numerator and denominator have no common factors?
3. How can we simplify more complex rational expressions involving polynomials of higher degrees?
4. What is the significance of factoring in solving rational equations?
5. How does factoring help in solving quadratic equations?

**Tip:** Always check if both the numerator and denominator can be factored completely before canceling common factors.

Fully simplify the fraction: (2x^2 + 11x - 40) / (x^2 + 5x - 24)

Learn how to fully simplify the rational expression (2x^2 + 11x - 40) / (x^2 + 5x - 24) by factoring the quadratic equations in both the numerator and denominator. A step-by-step guide on factoring and simplifying, suitable for students in Grades 9-11.

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