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To simplify the given expression:

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

we start by factoring both the numerator and the denominator.

**Factoring the numerator \(2x^2 + x - 3\):**

We need factors of \(2 \times (-3) = -6\) that add up to \(+1\). The factors \(3\) and \(-2\) work:
- \(2x^2 + 3x - 2x - 3 = (2x^2 + 3x) + (-2x - 3)\)
- \(x(2x + 3) - 1(2x + 3)\)
- \((x - 1)(2x + 3)\)

**Factoring the denominator \(2x^2 - 5x + 3\):**

We need factors of \(2 \times 3 = 6\) that add up to \(-5\). The factors \(-2\) and \(-3\) work:
- \(2x^2 - 2x - 3x + 3 = (2x^2 - 2x) + (-3x + 3)\)
- \(2x(x - 1) - 3(x - 1)\)
- \((2x - 3)(x - 1)\)

Now we substitute these factored forms back into the original expression:

\[
\frac{(x - 1)(2x + 3)}{(2x - 3)(x - 1)}
\]

We can cancel out the \((x - 1)\) terms in the numerator and the denominator (assuming \(x \neq 1\) to avoid division by zero):

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

This is the simplified form of the expression. Would you like to explore more details or any specific part of this simplification?

Further, here are some related topics you might find interesting:
1. How would this simplification change if one of the coefficients were different?
2. What happens if the numerator and denominator share more than one common factor?
3. Explore the graphical representation of the original and simplified expressions.
4. Discuss the implications of the restrictions \(x \neq 1\) in real-world scenarios.
5. Investigate how adding a constant to both the numerator and denominator affects the simplification process.

Tip: Always check for common factors in both the numerator and the denominator to simplify rational expressions, and remember to state any restrictions on the variable based on the factors you cancel.

Simplifique al máximo la siguiente expresión: (2x^2 + x - 3) / (2x^2 - 5x + 3)

Learn how to simplify the rational expression (2x^2 + x - 3) / (2x^2 - 5x + 3) by factoring both the numerator and denominator. This solution covers key algebraic concepts such as factoring quadratics and simplifying expressions, making it suitable for high school students.

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