To determine the correct numbers to place in the boxes for the expression [tex]\(x^2 + 3x - 18\)[/tex], let's follow a detailed step-by-step factorization process:
1. Identify the factors of the constant term (-18):
We need to find pairs of numbers that multiply to [tex]\(-18\)[/tex]. These pairs include:
[tex]\[
(-1, 18), (1, -18), (-2, 9), (2, -9), (-3, 6), (3, -6)
\][/tex]
2. Find the pair that sums to the coefficient of the [tex]\(x\)[/tex]-term (3):
We need a pair of numbers from the above list that adds up to 3. Let's check each pair:
- [tex]\((-1) + 18 = 17\)[/tex]
- [tex]\(1 + (-18) = -17\)[/tex]
- [tex]\((-2) + 9 = 7\)[/tex]
- [tex]\(2 + (-9) = -7\)[/tex]
- [tex]\((-3) + 6 = 3\)[/tex]
- [tex]\(3 + (-6) = -3\)[/tex]
The pair [tex]\((-3) + 6\)[/tex] adds up to 3.
3. Place the numbers in the binomial factors:
We have identified that [tex]\(-3\)[/tex] and [tex]\(6\)[/tex] are the numbers that multiply to [tex]\(-18\)[/tex] and add to 3. This means we can factorize the quadratic expression as:
[tex]\[
x^2 + 3x - 18 = (x + 6)(x - 3)
\][/tex]
Since the positioning of the numbers in the boxes is specified from left to right, the numbers that should be placed in the boxes are [tex]\(6\)[/tex] and [tex]\(-3\)[/tex].
Therefore, the correct numbers to place in the boxes from left to right are:
[tex]\[ \boxed{6 \text{ and } -3} \][/tex]
This concludes our detailed factorization process for the given quadratic expression.
