The problem asks us to factor the polynomial $2x^5 - 48x^3 - 50x$ completely, if possible. If the polynomial cannot be factored, we select the option that the polynomial is prime.

1. Problem Description

The problem asks us to factor the polynomial $2x^5 - 48x^3 - 50x$ completely, if possible. If the polynomial cannot be factored, we select the option that the polynomial is prime.

2. Solution Steps

First, we factor out the greatest common factor (GCF) from all the terms. The GCF of $2x^5$, $-48x^3$, and $-50x$ is $2x$.

$2x^5 - 48x^3 - 50x = 2x(x^4 - 24x^2 - 25)$

Now, we need to factor the quadratic-like expression $x^4 - 24x^2 - 25$. We can rewrite it as $(x^2)^2 - 24(x^2) - 25$. Let $y = x^2$. Then the expression becomes $y^2 - 24y - 25$.

We are looking for two numbers that multiply to $-25$ and add to $-24$. Those numbers are $-25$ and $1$.

Therefore, $y^2 - 24y - 25 = (y - 25)(y + 1)$.

Now, substitute $x^2$ back in for $y$:

$(x^2 - 25)(x^2 + 1)$

We recognize $x^2 - 25$ as a difference of squares, so we can factor it as $(x - 5)(x + 5)$. The term $x^2 + 1$ cannot be factored further using real numbers.

So, $x^4 - 24x^2 - 25 = (x - 5)(x + 5)(x^2 + 1)$.

Finally, we multiply by the GCF that we factored out earlier:

$2x(x^4 - 24x^2 - 25) = 2x(x - 5)(x + 5)(x^2 + 1)$

3. Final Answer

$2x(x-5)(x+5)(x^2+1)$