Answer :

We start with the expression

[tex]$$
x^5 - 25x^3.
$$[/tex]

Step 1. Factor out the Greatest Common Factor (GCF):

Both terms in the expression have a common factor of [tex]$x^3$[/tex], so we factor that out:

[tex]$$
x^5 - 25x^3 = x^3 \left(x^2 - 25\right).
$$[/tex]

Step 2. Factor the Remaining Expression:

The expression inside the parentheses,

[tex]$$
x^2 - 25,
$$[/tex]

is a difference of squares, which factors according to the formula

[tex]$$
a^2 - b^2 = (a + b)(a - b).
$$[/tex]

Here, [tex]$a = x$[/tex] and [tex]$b = 5$[/tex], so

[tex]$$
x^2 - 25 = (x + 5)(x - 5).
$$[/tex]

Step 3. Write the Fully Factored Form:

Substitute the factored form of [tex]$x^2 - 25$[/tex] back into the expression:

[tex]$$
x^5 - 25x^3 = x^3 \cdot (x + 5)(x - 5).
$$[/tex]

Thus, the final factored form is

[tex]$$
\boxed{x^3 (x - 5)(x + 5)}.
$$[/tex]