Question

Which of the following expressions can be compared to 3x2+15x−42x2−4
, and is also an equivalent form?(1 point)
Responses

(3x+21)(x−2)x+2
Start Fraction left parenthesis 3 x plus 21 right parenthesis left parenthesis x minus 2 right parenthesis over x plus 2 End Fraction

3(x+7)x+2
Start Fraction 3 left parenthesis x plus 7 right parenthesis over x plus 2 End Fraction

3(x+7)
3 left parenthesis x plus 7 right parenthesis

3(5x−12)−4

Answer 1

To compare the expression \( \frac{3x^2 + 15x - 42}{2x - 4} \) with the provided options, we need to simplify the original expression first.

1. **Factor the numerator and the denominator** of \( \frac{3x^2 + 15x - 42}{2x - 4} \):
- The numerator \( 3x^2 + 15x - 42 \) can be factored:
\[
3x^2 + 15x - 42 = 3(x^2 + 5x - 14) = 3(x + 7)(x - 2)
\]
- The denominator \( 2x - 4 \) can be factored:
\[
2x - 4 = 2(x - 2)
\]

2. **Now rewrite the entire expression**:
\[
\frac{3(x + 7)(x - 2)}{2(x - 2)}
\]
For \( x \neq 2 \), we can simplify this to:
\[
\frac{3(x + 7)}{2}
\]

3. **Now we need to compare it to the given choices**:
- **Option 1:** \( \frac{(3x + 21)(x - 2)}{x + 2} \)
- **Option 2:** \( \frac{3(x + 7)}{x + 2} \)
- **Option 3:** \( 3(x + 7) \)
- **Option 4:** \( 3(5x - 12) - 4 \)

4. **Analyzing the options:**
- **Option 1:** Not equivalent because the denominator differs.
- **Option 2:** Not equivalent because of the denominator.
- **Option 3:** Equivalent! It simplifies directly to \( \frac{3(x + 7)}{2} \).
- **Option 4:** Expanding it yields \( 15x - 36 \), which is not equivalent to \( \frac{3(x + 7)}{2} \).

Thus, the correct answer is:

**3(x + 7)**