When reducing the expression (3x^2 + 10x - 8)/(5x^2 + 19x - 4), what is the result?

To simplify the expression ((3x^2 + 10x - 8)/(5x^2 + 19x - 4)), we start by factoring both the numerator and the denominator.

For the numerator (3x^2 + 10x - 8), we need to find two numbers that multiply to (3 \cdot -8 = -24) and add up to (10). Those numbers are (12) and (-2). We can then break down the middle term:

[3x^2 + 12x - 2x - 8]

Now, we can regroup and factor by grouping:

[3x(x + 4) - 2(x + 4)]

This yields:

[(3x - 2)(x + 4)]

For the denominator (5x^2 + 19x - 4), we look for two numbers that multiply to (5 \cdot -4 = -20) and add up to (19). These numbers are (20) and (-1):

[5x^2 + 20x - x - 4]

We can now regroup