Answer :

We start with the expression

[tex]$$
\frac{x^7 - x^\beta}{4x^9 - 4x^5}.
$$[/tex]

Step 1. Factor the Denominator

Factor out the common term in the denominator:

[tex]$$
4x^9 - 4x^5 = 4x^5(x^4 - 1).
$$[/tex]

Notice that the term [tex]$x^4-1$[/tex] is a difference of squares. It can be factored further:

[tex]$$
x^4 - 1 = (x^2 - 1)(x^2 + 1) = (x - 1)(x + 1)(x^2 + 1).
$$[/tex]

Thus, the denominator may be written as

[tex]$$
4x^5(x-1)(x+1)(x^2+1).
$$[/tex]

Step 2. Factor the Numerator

The numerator is

[tex]$$
x^7 - x^\beta.
$$[/tex]

Since the exponent [tex]$\beta$[/tex] is not specified further, the numerator remains as

[tex]$$
x^7 - x^\beta.
$$[/tex]

Step 3. Write the Simplified Expression

After factoring, the expression becomes

[tex]$$
\frac{x^7 - x^\beta}{4x^5(x-1)(x+1)(x^2+1)}.
$$[/tex]

Alternatively, one may leave the denominator using the factorization [tex]$x^4 - 1$[/tex]:

[tex]$$
\frac{x^7 - x^\beta}{4x^5(x^4-1)}.
$$[/tex]

This is the simplified form of the original expression.

Final Answer

The original expression, its factorizations, and the simplified result are:

[tex]$$
\frac{x^7 - x^\beta}{4x^9-4x^5}, \quad x^7 - x^\beta, \quad 4x^5 (x-1)(x+1)(x^2+1), \quad \frac{x^7-x^\beta}{4x^5(x^4-1)}.
$$[/tex]

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Rewritten by : Barada