Factor -4x + 24 – 4x^2 completely. A) (x – 3)(x + 2) B) (x + 3)(x – 2) C) 4(x + 3)(x – 2) D) -4(x + 3)(x – 2
The Correct Answer and Explanation is:
To factor the expression −4x+24−4×2-4x + 24 – 4x^2−4x+24−4×2 completely, follow these steps:
Step 1: Rearrange the terms
First, rewrite the expression in standard form:−4x+24−4×2=−4×2−4x+24-4x + 24 – 4x^2 = -4x^2 – 4x + 24−4x+24−4×2=−4×2−4x+24
Step 2: Factor out the greatest common factor (GCF)
The GCF of all the terms is −4-4−4, so we factor −4-4−4 out of the entire expression:−4(x2+x−6)-4(x^2 + x – 6)−4(x2+x−6)
Step 3: Factor the quadratic expression
Now, we need to factor the quadratic expression x2+x−6x^2 + x – 6×2+x−6. We need to find two numbers that multiply to −6-6−6 and add to 111 (the coefficient of the middle term, xxx).
The two numbers that satisfy these conditions are 333 and −2-2−2, because:3×(−2)=−6and3+(−2)=13 \times (-2) = -6 \quad \text{and} \quad 3 + (-2) = 13×(−2)=−6and3+(−2)=1
Thus, we can factor x2+x−6x^2 + x – 6×2+x−6 as:(x+3)(x−2)(x + 3)(x – 2)(x+3)(x−2)
Step 4: Write the completely factored expression
Now substitute the factored form of the quadratic back into the expression:−4(x+3)(x−2)-4(x + 3)(x – 2)−4(x+3)(x−2)
Step 5: Final Answer
So, the completely factored form of the given expression is:−4(x+3)(x−2)\boxed{-4(x + 3)(x – 2)}−4(x+3)(x−2)
Explanation of the Answer
We first factored out the GCF, −4-4−4, and then factored the quadratic expression x2+x−6x^2 + x – 6×2+x−6 by finding two numbers that multiply to −6-6−6 and add to 111. This process gave us the factors (x+3)(x + 3)(x+3) and (x−2)(x – 2)(x−2), which we then combined with the GCF. Therefore, the correct answer is option D:−4(x+3)(x−2)\boxed{-4(x + 3)(x – 2)}−4(x+3)(x−2)
