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

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

• A. (3x - 2)/(5x - 1)
• B. (3x + 2)/(5x + 1)
• C. (3x)/(5x - 1)
• D. (3x - 3)/(5x + 4)

Answer: (A)

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

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