Question

Find the expression that is equivalent to 3x^2+2x−5x−2x^2+2
.(1 point)
Responses

A) (x+1)(x+2)
left parenthesis x plus 1 right parenthesis left parenthesis x plus 2 right parenthesis

B) (x−2)(x−1)
left parenthesis x minus 2 right parenthesis left parenthesis x minus 1 right parenthesis

C) (x+2)(x−2)
left parenthesis x plus 2 right parenthesis left parenthesis x minus 2 right parenthesis

D) (3x+2)(x+1)
left parenthesis 3 x plus 2 right parenthesis left parenthesis x plus 1 right parenthesis

Answer 1

To simplify the expression \(3x^2 + 2x - 5x - 2x^2 + 2\), we first combine like terms.

1. Combine the terms involving \(x^2\):
\[
3x^2 - 2x^2 = x^2
\]

2. Combine the terms involving \(x\):
\[
2x - 5x = -3x
\]

3. Combine the constant terms:
\[
2
\]

Putting it all together, we have:
\[
x^2 - 3x + 2
\]

Next, we will factor the quadratic \(x^2 - 3x + 2\). To factor it, we need to find two numbers that multiply to \(2\) (the constant term) and add to \(-3\) (the coefficient of \(x\)).

The numbers that satisfy these conditions are \(-1\) and \(-2\), since:
\[
-1 \cdot -2 = 2 \quad \text{and} \quad -1 + -2 = -3
\]

Thus, we can factor the expression as:
\[
(x - 1)(x - 2)
\]

Now, let's compare our factored result with the provided options:

- A) \((x + 1)(x + 2)\)
- B) \((x - 2)(x - 1)\)
- C) \((x + 2)(x - 2)\)
- D) \((3x + 2)(x + 1)\)

The correct option that matches our factorization \( (x - 1)(x - 2) \) is:

B) \((x - 2)(x - 1)\)

Thus, the answer is **B**.