Simplify :
` (x^(4) - 16) / (x^(3) + 2x^(2)+ 4x + 8 ) `

Answer

To simplify the expression \((x^{4} - 16) / (x^{3} + 2x^{2} + 4x + 8)\), we will follow these steps:

### Step 1: Factor the numerator
The numerator \(x^{4} - 16\) is a difference of squares. We can use the formula \(a^2 - b^2 = (a - b)(a + b)\).

Here, we can rewrite \(x^{4} - 16\) as:
\[ x^{4} - 16 = (x^{2})^{2} - (4)^{2} = (x^{2} - 4)(x^{2} + 4) \]

### Step 2: Factor \(x^{2} - 4\)
The term \(x^{2} - 4\) is also a difference of squares. We can factor it further:
\[ x^{2} - 4 = (x - 2)(x + 2) \]

Thus, the numerator can be expressed as:
\[ x^{4} - 16 = (x - 2)(x + 2)(x^{2} + 4) \]

### Step 3: Factor the denominator
Now, let's look at the denominator \(x^{3} + 2x^{2} + 4x + 8\). We can try to factor it by grouping.

Group the terms:
\[ (x^{3} + 2x^{2}) + (4x + 8) \]
Factoring out common terms from each group:
\[ x^{2}(x + 2) + 4(x + 2) \]
Now, we can factor out \((x + 2)\):
\[ (x + 2)(x^{2} + 4) \]

### Step 4: Rewrite the expression
Now we can rewrite the original expression with the factored forms:
\[ \frac{(x - 2)(x + 2)(x^{2} + 4)}{(x + 2)(x^{2} + 4)} \]

### Step 5: Cancel common factors
We can see that \((x + 2)\) and \((x^{2} + 4)\) are common in both the numerator and the denominator. Thus, we can cancel these terms:
\[ \frac{(x - 2) \cancel{(x + 2)} \cancel{(x^{2} + 4)}}{\cancel{(x + 2)} \cancel{(x^{2} + 4)}} \]
This simplifies to:
\[ x - 2 \]

### Final Answer
The simplified expression is:
\[ \boxed{x - 2} \]

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