Final answer:
The equation in factored form is already given. To write it in standard form, expand the numerator and denominator to get f(x) = (-6x² + 8x + 8) / (2x² + 7x + 5). The standard form is a single fraction with polynomials in the numerator and denominator.
Step-by-step explanation:
The factored form of the equation f(x) = -2(3x + 2)(x - 2) / (2x + 5)(x + 1) is already given. To write this as a standard form, we need to expand the numerator and denominator and then simplify if possible. As this is a rational function, the standard form would typically mean getting a single rational expression where the numerator and denominator are polynomials expressed as sums of terms in descending powers of x.
First, let's expand the numerator and denominator:
- Numerator: -2(3x + 2)(x - 2) = -2(3x² - 6x + 2x - 4) = -2(3x² - 4x - 4)
- Denominator: (2x + 5)(x + 1) = (2x² + 2x + 5x + 5) = (2x² + 7x + 5)
Therefore, the standard form of the equation is:
f(x) = (-6x² + 8x + 8) / (2x² + 7x + 5)
However, it might not be possible to simplify this fraction further without performing long division, which could result in a polynomial plus a remainder over the original denominator.