The polynomial [tex]$10x^3 + 35x^2 - 4x - 14$[/tex] is factored by grouping.

[tex]
\begin{array}{l}
10x^3 + 35x^2 - 4x - 14 \\
5x^2(2x + 7) - 2(2x + 7)
\end{array}
[/tex]

What is the common factor that is missing from both sets of parentheses?

A. [tex]-2x - 7[/tex]
B. [tex]2x + 7[/tex]
C. [tex]-2x^2 + 7[/tex]
D. [tex]2x^2 + 7[/tex]



Answer :

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].