Alright, let's walk through the solution step-by-step to simplify the given expression and check which option matches the simplified expression.
### Given Expression:
[tex]\[ 4x + 2 - \frac{x^2 + x - 12}{x - 3} \][/tex]
Step 1: Factorizing the Quadratic Expression
First, factorize the quadratic expression in the numerator of the fraction:
[tex]\[ x^2 + x - 12 \][/tex]
This can be factored as:
[tex]\[ x^2 + x - 12 = (x + 4)(x - 3) \][/tex]
So the expression now becomes:
[tex]\[ 4x + 2 - \frac{(x + 4)(x - 3)}{x - 3} \][/tex]
Step 2: Simplify the Fraction
Next, we can cancel the common factor [tex]\((x - 3)\)[/tex] in the numerator and denominator of the fraction:
[tex]\[ \frac{(x + 4)(x - 3)}{x - 3} = x + 4 \][/tex]
Now, substituting back into the original expression:
[tex]\[ 4x + 2 - (x + 4) \][/tex]
Step 3: Simplify the Subtraction
Let's distribute and combine like terms:
[tex]\[ 4x + 2 - x - 4 \][/tex]
[tex]\[ = 4x - x + 2 - 4 \][/tex]
[tex]\[ = 3x - 2 \][/tex]
### Simplified Expression:
The simplified expression is:
[tex]\[ 3x - 2 \][/tex]
### Matching the Options:
We need to find which option matches our simplified expression:
1. [tex]\(3x - 2\)[/tex]
2. [tex]\(3x + 4\)[/tex]
3. [tex]\(5x - 4\)[/tex]
4. [tex]\(5x + 4\)[/tex]
Clearly, the simplified expression [tex]\(3x - 2\)[/tex] matches the first option.
### Conclusion:
The correct simplified form of the given expression is:
[tex]\[ 3x - 2 \][/tex]
and the correct choice from the given options is the first one.
