`32x^4-500x`

Explanation

To factor the polynomial \(32x^4 - 500x\), we will follow these steps:

### Step 1: Factor out the greatest common factor (GCF)
First, we need to find the GCF of the terms \(32x^4\) and \(-500x\). The GCF here is \(4x\).

\[ 32x^4 - 500x = 4x(8x^3 - 125) \]

### Step 2: Recognize the difference of cubes
Next, we notice that \(8x^3 - 125\) can be expressed as a difference of cubes. We can rewrite it as:

\[ 8x^3 - 125 = (2x)^3 - 5^3 \]

### Step 3: Apply the difference of cubes formula
The difference of cubes can be factored using the formula:

\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]

In our case, \(a = 2x\) and \(b = 5\). Therefore, we can apply the formula:

\[ (2x - 5)((2x)^2 + (2x)(5) + (5)^2) \]

Calculating the second part:

\[ (2x)^2 = 4x^2, \quad (2x)(5) = 10x, \quad (5)^2 = 25 \]

So, we have:

\[ (2x)^2 + (2x)(5) + (5)^2 = 4x^2 + 10x + 25 \]

### Step 4: Combine everything
Now, substituting back into our expression, we get:

\[ 32x^4 - 500x = 4x(2x - 5)(4x^2 + 10x + 25) \]

### Final Answer
Thus, the factored form of \(32x^4 - 500x\) is:

\[ 4x(2x - 5)(4x^2 + 10x + 25) \]

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