To factor the polynomial [tex]\(10x^3 + 35x^2 - 4x - 14\)[/tex] by grouping, let's follow each step carefully:
1. Group the terms:
[tex]\[
(10x^3 + 35x^2) + (-4x - 14)
\][/tex]
2. Factor out the common factors in each group:
- For the first group, [tex]\(10x^3 + 35x^2\)[/tex]:
[tex]\[
10x^3 + 35x^2 = 5x^2(2x + 7)
\][/tex]
- For the second group, [tex]\(-4x - 14\)[/tex]:
[tex]\[
-4x - 14 = -2(2x + 7)
\][/tex]
3. Write it as a product of two terms:
[tex]\[
10x^3 + 35x^2 - 4x - 14 = 5x^2(2x + 7) - 2(2x + 7)
\][/tex]
4. Factor out the common binomial [tex]\((2x + 7)\)[/tex]:
[tex]\[
10x^3 + 35x^2 - 4x - 14 = (2x + 7)(5x^2 - 2)
\][/tex]
So, the common factor that appears in both sets of parentheses is [tex]\((2x + 7)\)[/tex]. Hence, the correct answer is:
[tex]\[
2x + 7
\][/tex]
