`(x+4)(x-4)(x^(2)+16)=` ____

Options

A. `x^(2)-64`
B. `x^(4)-64`
C. `x^(4)-256`
D. `x^(2)-256`

Answer

To solve the expression \((x+4)(x-4)(x^2+16)\), we can follow these steps:

### Step 1: Use the Difference of Squares Identity
The first part of the expression is \((x+4)(x-4)\). We can use the algebraic identity for the difference of squares:
\[ a^2 - b^2 = (a+b)(a-b) \]
Here, let \(a = x\) and \(b = 4\). Therefore:
\[ (x+4)(x-4) = x^2 - 4^2 = x^2 - 16 \]

### Step 2: Substitute Back into the Expression
Now, we can substitute this result back into the original expression:
\[ (x+4)(x-4)(x^2+16) = (x^2 - 16)(x^2 + 16) \]

### Step 3: Use the Difference of Squares Again
Now we have a new expression \((x^2 - 16)(x^2 + 16)\). We can apply the difference of squares identity again:
\[ a^2 - b^2 = (a+b)(a-b) \]
Here, let \(a = x^2\) and \(b = 16\). Therefore:
\[ (x^2 - 16)(x^2 + 16) = (x^2)^2 - 16^2 = x^4 - 256 \]

### Final Result
Thus, the expression \((x+4)(x-4)(x^2+16)\) simplifies to:
\[ \boxed{x^4 - 256} \]