To use the Distributive Property to rewrite the polynomial \((x+8)(x−2)(x+8)(x−2)\), we first notice that we can pair the factors:
\[ ((x + 8)(x - 2)) \times ((x + 8)(x - 2)). \]
First, we will simplify \((x + 8)(x - 2)\) using the Distributive Property (also known as the FOIL method for binomials).
\[ (x + 8)(x - 2) = x^2 - 2x + 8x - 16 = x^2 + 6x - 16. \]
Now, we have:
\[ ((x + 8)(x - 2))^2 = (x^2 + 6x - 16)(x^2 + 6x - 16). \]
Next, we use the Distributive Property again (or the FOIL method) to expand this:
\[ = (x^2 + 6x - 16)(x^2 + 6x - 16). \]
Using the Distributive Property:
- \(x^2 \cdot x^2 = x^4\)
- \(x^2 \cdot 6x = 6x^3\)
- \(x^2 \cdot -16 = -16x^2\)
- \(6x \cdot x^2 = 6x^3\)
- \(6x \cdot 6x = 36x^2\)
- \(6x \cdot -16 = -96x\)
- \(-16 \cdot x^2 = -16x^2\)
- \(-16 \cdot 6x = -96x\)
- \(-16 \cdot -16 = 256\)
Now, we combine all the terms together:
\[ x^4 + 6x^3 + 6x^3 - 16x^2 + 36x^2 - 96x - 96x + 256. \]
Combining like terms gives:
\[ x^4 + 12x^3 + (36 - 16 - 16)x^2 - 192x + 256 = x^4 + 12x^3 + 4x^2 - 192x + 256. \]
Thus, the expanded polynomial form is:
\[ x^4 + 12x^3 + 4x^2 - 192x + 256. \]
So the final answer is:
\[ x^4 + 12x^3 + 4x^2 - 192x + 256. \]