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Simplify the expression:

[tex]\[
\begin{array}{l}
4(5) + 2 - \frac{5^2 + 5 - 12}{5 - 3} \\
= 4(5) + 2 - \frac{25 + 5 - 12}{5 - 3} \\
= 4(5) + 2 - \frac{18}{2} \\
= 20 + 2 - 9 \\
= 13
\end{array}
\][/tex]

Which of the following simplifies the expression [tex]\( 4x + 2 - \frac{x^2 + x - 12}{x - 3} \)[/tex]?

A. [tex]\( 3x - 2 \)[/tex]

B. [tex]\( 3x + 4 \)[/tex]

C. [tex]\( 5x - 4 \)[/tex]

D. [tex]\( 5x + 4 \)[/tex]

Sagot :

GinnyAnsweridn
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.
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