To factor the polynomial [tex]\(10x^3 + 35x^2 - 4x - 14\)[/tex] by grouping, let's follow a step-by-step approach:
1. Group the terms in pairs:
[tex]\[
(10x^3 + 35x^2) - (4x + 14)
\][/tex]
2. Factor out the greatest common divisor (GCD) from each group:
- In the first group [tex]\(10x^3 + 35x^2\)[/tex], the GCD is [tex]\(5x^2\)[/tex]:
[tex]\[
10x^3 + 35x^2 = 5x^2(2x + 7)
\][/tex]
- In the second group [tex]\(-4x - 14\)[/tex], the GCD is [tex]\(-2\)[/tex]:
[tex]\[
-4x - 14 = -2(2x + 7)
\][/tex]
3. Rewrite the polynomial with the factored groups:
[tex]\[
5x^2(2x + 7) - 2(2x + 7)
\][/tex]
4. Identify the common factor in both terms:
- The common factor in both [tex]\(5x^2(2x + 7)\)[/tex] and [tex]\(-2(2x + 7)\)[/tex] is [tex]\((2x + 7)\)[/tex].
Finally, the common factor from both sets of parentheses is:
[tex]\[
\boxed{2x + 7}
\][/tex]
Therefore, the correct option is [tex]\(2x + 7\)[/tex].
