Answer :

To factor the expression [tex]\(35x^3 - 20x^2 + 7x - 4\)[/tex] by grouping, we can follow these steps:

1. Group the terms:
We start by grouping the terms into two pairs. Here, the expression is:

[tex]\[
35x^3 - 20x^2 + 7x - 4
\][/tex]

We can group it like this:

[tex]\[
(35x^3 - 20x^2) + (7x - 4)
\][/tex]

2. Factor each group separately:
- First group: [tex]\(35x^3 - 20x^2\)[/tex]

In this group, we can factor out the greatest common factor, which is [tex]\(5x^2\)[/tex]:

[tex]\[
35x^3 - 20x^2 = 5x^2(7x - 4)
\][/tex]

- Second group: [tex]\(7x - 4\)[/tex]

This group doesn't have any common factor other than 1, so it remains:

[tex]\[
7x - 4
\][/tex]

3. Identify the common binomial factor:
Notice that both factored groups have the common factor [tex]\((7x - 4)\)[/tex].

4. Factor out the common binomial:
Now, factor [tex]\((7x - 4)\)[/tex] out of the entire expression:

[tex]\[
35x^3 - 20x^2 + 7x - 4 = (7x - 4)(5x^2 + 1)
\][/tex]

Thus, the original expression factors to:

[tex]\[
(7x - 4)(5x^2 + 1)
\][/tex]

And that's the final factored form of the expression!

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