Question

1. $k^2 + 17k + 66$

Explanation:

Step1: Find two numbers that multiply to 66 and add to 17.

We need two numbers \( m \) and \( n \) such that \( m \times n = 66 \) and \( m + n = 17 \). The factors of 66 are: 1 & 66, 2 & 33, 3 & 22, 6 & 11. Among these, 6 and 11 add up to 17 (\( 6 + 11 = 17 \)) and multiply to 66 (\( 6 \times 11 = 66 \)).

Step2: Factor the quadratic expression.

Using the numbers 6 and 11, we can rewrite the middle term of the quadratic \( k^2 + 17k + 66 \) as \( 6k + 11k \). So,
\[

$$\begin{align*} k^2 + 17k + 66 &= k^2 + 6k + 11k + 66 \\ &= (k^2 + 6k) + (11k + 66) \\ &= k(k + 6) + 11(k + 6) \\ &= (k + 6)(k + 11) \end{align*}$$

\]

Answer:

\( (k + 6)(k + 11) \)