The common factor of `(p^(2) + 9p + 14)` and `(p^(2) + 13p + 42)` is .

Options

A. p+2
B. p+6
C. p+3
D. p+7

Answer

To find the common factor of the two expressions \( p^2 + 9p + 14 \) and \( p^2 + 13p + 42 \), we will follow these steps:

### Step 1: Factor the first expression \( p^2 + 9p + 14 \)

We need to factor \( p^2 + 9p + 14 \). We look for two numbers that multiply to \( 14 \) (the constant term) and add up to \( 9 \) (the coefficient of \( p \)).

The numbers \( 7 \) and \( 2 \) satisfy these conditions:
- \( 7 \times 2 = 14 \)
- \( 7 + 2 = 9 \)

Thus, we can write:
\[ p^2 + 9p + 14 = (p + 7)(p + 2) \]

### Step 2: Factor the second expression \( p^2 + 13p + 42 \)

Next, we factor \( p^2 + 13p + 42 \). We need two numbers that multiply to \( 42 \) and add up to \( 13 \).

The numbers \( 6 \) and \( 7 \) satisfy these conditions:
- \( 6 \times 7 = 42 \)
- \( 6 + 7 = 13 \)

Thus, we can write:
\[ p^2 + 13p + 42 = (p + 6)(p + 7) \]

### Step 3: Identify the common factor

Now, we have the factorizations:
- \( p^2 + 9p + 14 = (p + 7)(p + 2) \)
- \( p^2 + 13p + 42 = (p + 6)(p + 7) \)

The common factor in both expressions is \( p + 7 \).

### Conclusion

The common factor of \( p^2 + 9p + 14 \) and \( p^2 + 13p + 42 \) is:
\[ \boxed{p + 7} \]