Answer :

We start with the polynomial

[tex]$$
4x^3 + 20x^2 - 9x - 45.
$$[/tex]

Our goal is to factor it completely. One effective technique is factoring by grouping.

1. Group the terms:

Group the first two terms together and the last two terms together:

[tex]$$
(4x^3 + 20x^2) + (-9x - 45).
$$[/tex]

2. Factor each group:

- In the first group, factor out the common factor [tex]$4x^2$[/tex]:

[tex]$$
4x^3 + 20x^2 = 4x^2(x + 5).
$$[/tex]

- In the second group, factor out the common factor [tex]$-9$[/tex]:

[tex]$$
-9x - 45 = -9(x + 5).
$$[/tex]

3. Factor out the common binomial:

Notice that both groups contain the common factor [tex]$(x + 5)$[/tex]. Factor [tex]$(x + 5)$[/tex] out:

[tex]$$
4x^2(x + 5) - 9(x + 5) = (x + 5)\left(4x^2 - 9\right).
$$[/tex]

4. Factor the quadratic expression:

The expression inside the parentheses, [tex]$4x^2 - 9$[/tex], is a difference of squares since

[tex]$$
4x^2 = (2x)^2 \quad \text{and} \quad 9 = 3^2.
$$[/tex]

Hence, using the difference of squares formula

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

we factor [tex]$4x^2 - 9$[/tex] as

[tex]$$
4x^2 - 9 = (2x - 3)(2x + 3).
$$[/tex]

5. Write the final factorization:

Combining the factors, the complete factorization of the polynomial is

[tex]$$
4x^3 + 20x^2 - 9x - 45 = (x + 5)(2x - 3)(2x + 3).
$$[/tex]

This is the fully factored form of the given polynomial.

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