To simplify the given rational expression \( \frac{6(x^2 - 4x) + 3(2x - 8)}{2(x^2 - 2x) + (-2x - 8)} \), we will first simplify the numerator and the denominator separately.
Step 1: Simplify the numerator
-
Expand \( 6(x^2 - 4x) \): \[ 6x^2 - 24x \]
-
Simplify \( 3(2x - 8) \): \[ 6x - 24 \]
-
Combine the results for the numerator: \[ 6x^2 - 24x + 6x - 24 = 6x^2 - 18x - 24 \]
Step 2: Simplify the denominator
-
Expand \( 2(x^2 - 2x) \): \[ 2x^2 - 4x \]
-
Combine it with \( -2x - 8 \): \[ 2x^2 - 4x - 2x - 8 = 2x^2 - 6x - 8 \]
Step 3: Combine the simplified results
Now we have the rational expression as: \[ \frac{6x^2 - 18x - 24}{2x^2 - 6x - 8} \]
Step 4: Factor both parts if possible
-
For the numerator \( 6x^2 - 18x - 24 \): \[ = 6(x^2 - 3x - 4) = 6(x - 4)(x + 1) \quad \text{(factoring the quadratic)} \]
-
For the denominator \( 2x^2 - 6x - 8 \): \[ = 2(x^2 - 3x - 4) = 2(x - 4)(x + 1) \quad \text{(factoring the quadratic)} \]
Step 5: Simplify the overall expression
Now, the rational expression becomes: \[ \frac{6(x - 4)(x + 1)}{2(x - 4)(x + 1)} \]
Since \( (x - 4)(x + 1) \) cancels out: \[ = \frac{6}{2} = 3 \]
Thus, the simplified equivalent form is: \[ 3 \]
Therefore, the correct response is 3.