To find the product of [tex]\((2x + 4)(4x - 1)(x + 4)\)[/tex], we will expand the expression step-by-step.
Step 1: Expand the first two terms [tex]\((2x + 4)(4x - 1)\)[/tex].
Let's calculate this product first:
[tex]\[
(2x + 4)(4x - 1)
\][/tex]
Use the distributive property (also known as the FOIL method for binomials):
1. [tex]\(2x \cdot 4x = 8x^2\)[/tex]
2. [tex]\(2x \cdot (-1) = -2x\)[/tex]
3. [tex]\(4 \cdot 4x = 16x\)[/tex]
4. [tex]\(4 \cdot (-1) = -4\)[/tex]
Combine these results:
[tex]\[
8x^2 - 2x + 16x - 4
\][/tex]
Simplify the combined terms:
[tex]\[
8x^2 + 14x - 4
\][/tex]
So, we have:
[tex]\[
(2x + 4)(4x - 1) = 8x^2 + 14x - 4
\][/tex]
Step 2: Multiply the resulting expression by the third term [tex]\((x + 4)\)[/tex].
Now, we will expand [tex]\((8x^2 + 14x - 4)(x + 4)\)[/tex]:
Distribute each term in [tex]\(8x^2 + 14x - 4\)[/tex] across [tex]\((x + 4)\)[/tex].
[tex]\[
8x^2 \cdot x + 8x^2 \cdot 4 + 14x \cdot x + 14x \cdot 4 - 4 \cdot x - 4 \cdot 4
\][/tex]
Calculate each term:
1. [tex]\(8x^2 \cdot x = 8x^3\)[/tex]
2. [tex]\(8x^2 \cdot 4 = 32x^2\)[/tex]
3. [tex]\(14x \cdot x = 14x^2\)[/tex]
4. [tex]\(14x \cdot 4 = 56x\)[/tex]
5. [tex]\((-4) \cdot x = -4x\)[/tex]
6. [tex]\((-4) \cdot 4 = -16\)[/tex]
Now, combine all these terms:
[tex]\[
8x^3 + 32x^2 + 14x^2 + 56x - 4x - 16
\][/tex]
Combine like terms:
[tex]\[
8x^3 + (32x^2 + 14x^2) + (56x - 4x) - 16
\][/tex]
Simplify:
[tex]\[
8x^3 + 46x^2 + 52x - 16
\][/tex]
Final Answer:
The expanded form of [tex]\((2x + 4)(4x - 1)(x + 4)\)[/tex] is:
[tex]\[
8x^3 + 46x^2 + 52x - 16
\][/tex]
