Sure, let's walk through the process of factoring the quadratic expression [tex]\( x^2 + 3x - 18 \)[/tex].
### Step 1: Understanding the Problem
We need to factor the expression into the form:
[tex]\[ (x + a)(x - b) \][/tex]
where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are numbers that we need to determine.
### Step 2: Setting Up the Factoring
The given expression is [tex]\( x^2 + 3x - 18 \)[/tex]. To factor this, we need to find two numbers whose product is equal to the constant term (-18) and whose sum is equal to the coefficient of the middle term (3).
### Step 3: Finding the Correct Pair of Factors
We need to find two numbers that multiply to -18 and add to 3. Let’s identify the pairs of factors of -18 and check their sums:
- (-1, 18): Their sum is [tex]\(-1 + 18 = 17\)[/tex]
- (1, -18): Their sum is [tex]\(1 - 18 = -17\)[/tex]
- (-2, 9): Their sum is [tex]\(-2 + 9 = 7\)[/tex]
- (2, -9): Their sum is [tex]\(2 - 9 = -7\)[/tex]
- (-3, 6): Their sum is [tex]\(-3 + 6 = 3\)[/tex]
- (3, -6): Their sum is [tex]\(3 - 6 = -3\)[/tex]
Among these pairs, the pair (-3, 6) sums up to 3.
### Step 4: Factoring the Expression
Given our correct pair [tex]\((6, -3)\)[/tex], we can write the factored form of the quadratic expression as:
[tex]\[ x^2 + 3x - 18 = (x + 6)(x - 3) \][/tex]
### Step 5: Conclusion
Thus, the numbers that should be placed in the boxes from left to right are:
[tex]\[
\boxed{6}
\quad \text{and} \quad
\boxed{3}
\][/tex]
Therefore, the correct answer is:
[tex]\[ 6 \text{ and } 3 \][/tex]
So, the factorized form of the expression [tex]\( x^2 + 3x - 18 \)[/tex] is [tex]\((x + 6)(x - 3)\)[/tex].
