Answer :
Absolutely! Let's work through the problem step-by-step to simplify the given expression: [tex]\(\frac{x^2 - 3x - 18}{x + 3}\)[/tex].
### Step 1: Factor the Numerator
First, we need to factor the numerator [tex]\(x^2 - 3x - 18\)[/tex].
To factor [tex]\(x^2 - 3x - 18\)[/tex], we look for two numbers that multiply to [tex]\(-18\)[/tex] (the constant term) and add to [tex]\(-3\)[/tex] (the coefficient of the [tex]\(x\)[/tex] term).
After considering pairs of factors of [tex]\(-18\)[/tex], we find that [tex]\(-6\)[/tex] and [tex]\(3\)[/tex] work since:
[tex]\[ -6 \times 3 = -18 \quad \text{and} \quad -6 + 3 = -3 \][/tex]
So, the numerator [tex]\(x^2 - 3x - 18\)[/tex] can be factored as:
[tex]\[ x^2 - 3x - 18 = (x - 6)(x + 3) \][/tex]
### Step 2: Express the Simplified Form
Now, substitute the factored form back into the expression:
[tex]\[ \frac{x^2 - 3x - 18}{x + 3} = \frac{(x - 6)(x + 3)}{x + 3} \][/tex]
### Step 3: Simplify the Expression
Next, we notice that the factor [tex]\(x + 3\)[/tex] in the numerator and the denominator can be canceled out:
[tex]\[ \frac{(x - 6)(x + 3)}{x + 3} = x - 6 \][/tex]
### Step 4: Consider the Domain
Lastly, we need to consider any restrictions on the domain. The original denominator [tex]\(x + 3\)[/tex] cannot be zero, thus:
[tex]\[ x + 3 \neq 0 \implies x \neq -3 \][/tex]
### Final Result:
So, the simplified expression is:
[tex]\[ x - 6, \quad \text{where } x \neq -3 \][/tex]
Therefore, among the given choices, the correct answer is:
[tex]\[ \boxed{x - 6, \text{ where } x \neq -3} \][/tex]
### Step 1: Factor the Numerator
First, we need to factor the numerator [tex]\(x^2 - 3x - 18\)[/tex].
To factor [tex]\(x^2 - 3x - 18\)[/tex], we look for two numbers that multiply to [tex]\(-18\)[/tex] (the constant term) and add to [tex]\(-3\)[/tex] (the coefficient of the [tex]\(x\)[/tex] term).
After considering pairs of factors of [tex]\(-18\)[/tex], we find that [tex]\(-6\)[/tex] and [tex]\(3\)[/tex] work since:
[tex]\[ -6 \times 3 = -18 \quad \text{and} \quad -6 + 3 = -3 \][/tex]
So, the numerator [tex]\(x^2 - 3x - 18\)[/tex] can be factored as:
[tex]\[ x^2 - 3x - 18 = (x - 6)(x + 3) \][/tex]
### Step 2: Express the Simplified Form
Now, substitute the factored form back into the expression:
[tex]\[ \frac{x^2 - 3x - 18}{x + 3} = \frac{(x - 6)(x + 3)}{x + 3} \][/tex]
### Step 3: Simplify the Expression
Next, we notice that the factor [tex]\(x + 3\)[/tex] in the numerator and the denominator can be canceled out:
[tex]\[ \frac{(x - 6)(x + 3)}{x + 3} = x - 6 \][/tex]
### Step 4: Consider the Domain
Lastly, we need to consider any restrictions on the domain. The original denominator [tex]\(x + 3\)[/tex] cannot be zero, thus:
[tex]\[ x + 3 \neq 0 \implies x \neq -3 \][/tex]
### Final Result:
So, the simplified expression is:
[tex]\[ x - 6, \quad \text{where } x \neq -3 \][/tex]
Therefore, among the given choices, the correct answer is:
[tex]\[ \boxed{x - 6, \text{ where } x \neq -3} \][/tex]
