To completely factor the quadratic expression [tex]\(4x^2 + 12x - 72\)[/tex], let's follow the steps to factor it step-by-step.
### Step 1: Identify Common Factors
First, we look for common factors in the terms of the quadratic expression. We notice that each term can be divided by 4:
[tex]\[ 4x^2 + 12x - 72 = 4(x^2 + 3x - 18) \][/tex]
### Step 2: Factor the Quadratic Expression Inside the Parentheses
Next, we focus on factoring the quadratic expression inside the parentheses: [tex]\(x^2 + 3x - 18\)[/tex].
We search for two numbers that multiply to [tex]\(-18\)[/tex] (the constant term) and add up to [tex]\(3\)[/tex] (the coefficient of the [tex]\(x\)[/tex] term).
The numbers that satisfy these conditions are [tex]\(6\)[/tex] and [tex]\(-3\)[/tex] because:
[tex]\[ 6 \times (-3) = -18 \][/tex]
[tex]\[ 6 + (-3) = 3 \][/tex]
This allows us to split the middle term and factor by grouping:
[tex]\[ x^2 + 3x - 18 = x^2 + 6x - 3x - 18 \][/tex]
[tex]\[ = x(x + 6) - 3(x + 6) \][/tex]
[tex]\[ = (x - 3)(x + 6) \][/tex]
### Step 3: Combine the Common Factor from Step 1
We put everything together by multiplying the common factor from Step 1 with the factored expression:
[tex]\[ 4(x^2 + 3x - 18) = 4(x - 3)(x + 6) \][/tex]
### Final Factored Expression
Thus, the completely factored form of the given quadratic expression [tex]\(4x^2 + 12x - 72\)[/tex] is:
[tex]\[ 4(x - 3)(x + 6) \][/tex]
So, the correct placement of terms for the expression is:
[tex]\[ 4(x - 3)(x + 6) \][/tex]
