Which of the following expressions can be compared to 3x2+15x−42x2−4

Question

, 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)