Answer :

Sure! Let's factor the expression [tex]\(30x^3 - 3 + 45x^2 - 2x\)[/tex] completely. We'll do this step-by-step.

1. Rearrange Terms:
Start by arranging the terms in descending order of the powers of [tex]\(x\)[/tex]:
[tex]\[
30x^3 + 45x^2 - 2x - 3
\][/tex]

2. Group Terms:
Group the terms in pairs to make factoring easier:
[tex]\[
(30x^3 + 45x^2) + (-2x - 3)
\][/tex]

3. Factor Each Pair:
- First Group ([tex]\(30x^3 + 45x^2\)[/tex]):
Factor out the greatest common factor (GCF), which is [tex]\(15x^2\)[/tex]:
[tex]\[
15x^2(2x + 3)
\][/tex]

- Second Group ([tex]\(-2x - 3\)[/tex]):
Factor out the negative GCF, which is [tex]\(-1\)[/tex]:
[tex]\[
-1(2x + 3)
\][/tex]

4. Combine Using Common Factors:
Notice we have a common binomial factor [tex]\((2x + 3)\)[/tex]. Combine the expression:
[tex]\[
15x^2(2x + 3) - 1(2x + 3) = (15x^2 - 1)(2x + 3)
\][/tex]

5. Write the Complete Factored Form:
The completely factored form of the expression is:
[tex]\[
(15x^2 - 1)(2x + 3)
\][/tex]

This is the factored form of the given polynomial: [tex]\( (15x^2 - 1)(2x + 3) \)[/tex].

If needed, you can further check if [tex]\(15x^2 - 1\)[/tex] can be factored. Since it follows the pattern of a difference of squares, you can write:
[tex]\[
15x^2 - 1 = ( \sqrt{15}x - 1)( \sqrt{15}x + 1)
\][/tex]
So the expression factors completely to:
[tex]\[
( \sqrt{15}x - 1)( \sqrt{15}x + 1)(2x + 3)
\][/tex]

However, depending on the context, sometimes it's preferred to keep the factored expression in terms of integers, so:
[tex]\[
(15x^2 - 1)(2x + 3)
\][/tex]
might be considered "completely" factored without utilizing the difference of squares.

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