Answer :

We want to factor the polynomial

[tex]$$24x^3 - 18x^2 - 60x + 45$$[/tex]

completely. Follow these steps:

1. Group the terms:

Group the polynomial into two pairs:

[tex]$$
(24x^3 - 18x^2) + (-60x + 45)
$$[/tex]

2. Factor each group:

- In the first group, [tex]\(24x^3 - 18x^2\)[/tex], the common factor is [tex]\(6x^2\)[/tex]:

[tex]$$
24x^3 - 18x^2 = 6x^2(4x - 3)
$$[/tex]

- In the second group, [tex]\(-60x + 45\)[/tex], the common factor is [tex]\(-15\)[/tex]:

[tex]$$
-60x + 45 = -15(4x - 3)
$$[/tex]

3. Factor out the common binomial:

Both groups now contain the factor [tex]\((4x-3)\)[/tex]. Factor [tex]\((4x-3)\)[/tex] from the entire expression:

[tex]$$
6x^2(4x-3) - 15(4x-3) = (4x-3)(6x^2 - 15)
$$[/tex]

4. Factor the quadratic expression:

Look at [tex]\(6x^2 - 15\)[/tex]. Notice that 3 is a common factor:

[tex]$$
6x^2 - 15 = 3(2x^2 - 5)
$$[/tex]

5. Write the complete factorization:

Substitute the factored form back into the expression:

[tex]$$
(4x-3)(6x^2 - 15) = (4x-3) \cdot 3(2x^2-5)
$$[/tex]

Rearranging slightly, the complete factorization is:

[tex]$$
3(4x-3)(2x^2-5)
$$[/tex]

Thus, the fully factored form of the polynomial is

[tex]$$\boxed{3(4x-3)(2x^2-5)}.$$[/tex]

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