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------------------------------------------------ Factor the trinomial completely.

[tex]x^9 - 4x^8 - 45x^7[/tex]

To factor the trinomial [tex]\(x^9 - 4x^8 - 45x^7\)[/tex] completely, you can follow these steps:

1. Factor Out the Greatest Common Factor (GCF):
First, observe that each term in the trinomial has a common factor of [tex]\(x^7\)[/tex]. We can factor [tex]\(x^7\)[/tex] out from the expression:
[tex]\[
x^9 - 4x^8 - 45x^7 = x^7(x^2 - 4x - 45)
\][/tex]

2. Factor the Quadratic Expression:
Now, focus on factoring the quadratic expression [tex]\(x^2 - 4x - 45\)[/tex]. We aim to find two numbers that multiply to [tex]\(-45\)[/tex] (the constant term) and add to [tex]\(-4\)[/tex] (the coefficient of the linear term).

- The pair of numbers that fits this requirement is [tex]\(-9\)[/tex] and [tex]\(5\)[/tex], because:
- [tex]\((-9) \times 5 = -45\)[/tex]
- [tex]\((-9) + 5 = -4\)[/tex]

Therefore, the quadratic expression can be factored as:
[tex]\[
x^2 - 4x - 45 = (x - 9)(x + 5)
\][/tex]

3. Combine the Factors:
Finally, we combine the factors together with the GCF we pulled out:
[tex]\[
x^9 - 4x^8 - 45x^7 = x^7(x - 9)(x + 5)
\][/tex]

So, the complete factorization of the trinomial [tex]\(x^9 - 4x^8 - 45x^7\)[/tex] is:
[tex]\[
x^7(x - 9)(x + 5)
\][/tex]