## Simplifying the Rational Expression: (6x² - 7x - 5) / (2x + 1)

This article will guide you through the process of simplifying the rational expression (6x² - 7x - 5) / (2x + 1).

### Understanding Rational Expressions

A rational expression is simply a fraction where the numerator and denominator are polynomials. To simplify a rational expression, we aim to factor both the numerator and denominator and cancel out any common factors.

### Factoring the Numerator

**Find two numbers that multiply to give -30 (the product of the leading coefficient and the constant term) and add up to -7 (the coefficient of the middle term).**These numbers are -10 and 3.**Rewrite the middle term (-7x) as the sum of these two numbers.**This gives us: 6x² - 10x + 3x - 5.**Factor by grouping:**- Group the first two terms and the last two terms: (6x² - 10x) + (3x - 5).
- Factor out the greatest common factor (GCF) from each group: 2x(3x - 5) + 1(3x - 5).
- Notice that both terms now have a common factor of (3x - 5). Factor this out: (3x - 5)(2x + 1).

### Simplifying the Expression

Now we have: (6x² - 7x - 5) / (2x + 1) = (3x - 5)(2x + 1) / (2x + 1)

Since (2x + 1) appears in both the numerator and denominator, we can cancel them out.

This leaves us with the simplified expression: **3x - 5**

### Conclusion

Therefore, the simplified form of the rational expression (6x² - 7x - 5) / (2x + 1) is **3x - 5**, provided that x ≠ -1/2 (as this would make the denominator zero). Remember, simplifying rational expressions often involves factoring and canceling out common factors to reach the most concise form.