Answer :

To factor the polynomial [tex]\(10x^3 - 25x^2 - 2x + 5\)[/tex] by grouping, we can follow these steps:

1. Group the terms: We want to rearrange and group the terms such that we can factor out a common factor from each group. The polynomial is:

[tex]\[
10x^3 - 25x^2 - 2x + 5
\][/tex]

Let's group it as follows:

[tex]\[
(10x^3 - 25x^2) + (-2x + 5)
\][/tex]

2. Factor out the common factor from each group:

- From the first group [tex]\(10x^3 - 25x^2\)[/tex], we can factor out [tex]\(5x^2\)[/tex]:

[tex]\[
5x^2(2x - 5)
\][/tex]

- From the second group [tex]\(-2x + 5\)[/tex], we factor out [tex]\(-1\)[/tex]:

[tex]\[
-1(2x - 5)
\][/tex]

3. Combine the factored groups: Now, notice that both groups have a common factor of [tex]\((2x - 5)\)[/tex]. We can combine the groups as follows:

[tex]\[
5x^2(2x - 5) - 1(2x - 5)
\][/tex]

This becomes:

[tex]\[
(5x^2 - 1)(2x - 5)
\][/tex]

So, the polynomial [tex]\(10x^3 - 25x^2 - 2x + 5\)[/tex] is factored as:

[tex]\[
(5x^2 - 1)(2x - 5)
\][/tex]

And that is the factorization of the given polynomial by grouping.

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