To factor the polynomial [tex]\(10x^3 + 35x^2 - 4x - 14\)[/tex] by grouping, follow these steps:
1. Group the terms:
Separate the polynomial into two groups:
[tex]\[
(10x^3 + 35x^2) + (-4x - 14)
\][/tex]
2. Factor out the Greatest Common Factor (GCF) from each group:
- For the first group [tex]\((10x^3 + 35x^2)\)[/tex], the GCF is [tex]\(5x^2\)[/tex]:
[tex]\[
10x^3 + 35x^2 = 5x^2(2x + 7)
\][/tex]
- For the second group [tex]\((-4x - 14)\)[/tex], the GCF is [tex]\(-2\)[/tex]:
[tex]\[
-4x - 14 = -2(2x + 7)
\][/tex]
3. Rewrite the polynomial with the factored terms:
[tex]\[
10x^3 + 35x^2 - 4x - 14 = 5x^2(2x + 7) - 2(2x + 7)
\][/tex]
4. Identify the common factor in both terms:
Both terms contain the common binomial factor [tex]\((2x + 7)\)[/tex].
Therefore, the correct missing factor in both sets of parentheses is:
[tex]\[
\boxed{2x + 7}
\][/tex]
