Factor completely. Remember to look first for a common factor and to check by multiplying.

[tex]-x^5 + 10x^4 - 25x^3[/tex]

Select the correct choice below and, if necessary, fill in the answer box to complete your choice:

A. [tex]-x^5 + 10x^4 - 25x^3 =[/tex] [tex]\square[/tex]

B. The polynomial is prime.

To factor the polynomial [tex]$-x^5 + 10x^4 - 25x^3$[/tex] completely, we can follow these steps:

1. Look for a common factor:
- Each term in the polynomial [tex]$-x^5$[/tex], [tex]$10x^4$[/tex], and [tex]$-25x^3$[/tex] includes a factor of [tex]$x^3$[/tex].
- Additionally, the coefficients [tex]$-1$[/tex], [tex]$10$[/tex], and [tex]$-25$[/tex] do not have any common factors other than 1.
- So, the greatest common factor (GCF) we can factor out is [tex]$-x^3$[/tex].

2. Factor out the GCF:
- When we factor [tex]$-x^3$[/tex] out of each term, the expression becomes:
[tex]\[
-x^3(x^2 - 10x + 25)
\][/tex]

3. Factor the remaining quadratic:
- Now, we need to factor the quadratic [tex]$x^2 - 10x + 25$[/tex].
- Notice that this quadratic is a perfect square trinomial. It can be rewritten as:
[tex]\[
(x - 5)(x - 5) \quad \text{or} \quad (x - 5)^2
\][/tex]

4. Write the complete factorization:
- Combine the factored quadratic with the factored out GCF:
[tex]\[
-x^3(x - 5)^2
\][/tex]

So, the complete factorization of the polynomial [tex]$-x^5 + 10x^4 - 25x^3$[/tex] is [tex]$-x^3(x - 5)^2$[/tex].

The correct choice is:
A. [tex]$-x^5 + 10x^4 - 25x^3 = -x^3(x - 5)^2$[/tex]