Sure! Let's simplify the given expression step-by-step.
The expression to simplify is:
[tex]\[
(2x - 4)(x + 4)
\][/tex]
### Step 1: Apply the Distributive Property
First, we'll apply the distributive property to expand the expression. The distributive property states that [tex]\(a(b+c) = ab + ac\)[/tex]. Here, we will distribute each term of the first polynomial to each term in the second polynomial.
[tex]\[
(2x - 4)(x + 4) = 2x(x) + 2x(4) - 4(x) - 4(4)
\][/tex]
### Step 2: Multiply Each Term
Now we'll multiply each pair of terms:
[tex]\[
= (2x \cdot x) + (2x \cdot 4) - (4 \cdot x) - (4 \cdot 4)
\][/tex]
[tex]\[
= 2x^2 + 8x - 4x - 16
\][/tex]
### Step 3: Combine Like Terms
Next, we'll combine the like terms:
[tex]\[
2x^2 + (8x - 4x) - 16
\][/tex]
[tex]\[
= 2x^2 + 4x - 16
\][/tex]
### Step 4: Factor the Expression
Finally, we'll factor the simplified expression. Notice that [tex]\(2x^2 + 4x - 16\)[/tex] can be factored further. Let’s factor out the greatest common factor (GCF) first, which is 2:
[tex]\[
= 2(x^2 + 2x - 8)
\][/tex]
Now we need to factor the quadratic [tex]\(x^2 + 2x - 8\)[/tex]. We look for two numbers that multiply to [tex]\(-8\)[/tex] and add to [tex]\(2\)[/tex]. Those numbers are [tex]\(4\)[/tex] and [tex]\(-2\)[/tex]:
[tex]\[
= 2(x + 4)(x - 2)
\][/tex]
Therefore, the simplified expression is:
[tex]\[
2(x - 2)(x + 4)
\][/tex]
Thus, the result of simplifying the expression [tex]\((2x - 4)(x + 4)\)[/tex] is:
[tex]\[
2(x - 2)(x + 4)
\][/tex]
