## To simplify the function f(x) = 4x^2 - 12x / x^2 + x - 12, you can follow these steps:

1. Factor the numerator, 4x^2 - 12x:

The greatest common factor of 4x^2 and -12x is 4x, so you can rewrite the numerator as 4x(x - 3).

2. Factor the denominator, x^2 + x - 12:

To factor a quadratic expression, you need to look for two numbers that multiply to give the last term (-12) and add up to give the coefficient of the middle term (1 in this case).

The numbers that satisfy these conditions are 4 and -3, so you can rewrite the denominator as (x + 4)(x - 3).

3. Cancel out common factors:

Now that the numerator and denominator are factored, you can cancel out the common factor of (x - 3). This leaves you with:

f(x) = 4x / (x + 4).

That's the simplified form of the function.

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