Question

Find the expression that is equivalent to 3𝑥2+2𝑥−5𝑥−2𝑥2+2 (𝑥−2)(𝑥−1) (𝑥+2)(𝑥−2) (𝑥+1)(𝑥+2)

Answer 1

To simplify the given expression \( \frac{3x^2 + 2x - 5x - 2x^2 + 2}{(x - 2)(x - 1)(x + 2)(x - 2)(x + 1)(x + 2)} \), we first need to simplify the numerator and the denominator.

### Step 1: Simplify the Numerator

Start with the expression in the numerator:

\[
3x^2 + 2x - 5x - 2x^2 + 2
\]

Combine like terms:

\[
(3x^2 - 2x^2) + (2x - 5x) + 2
\]

This results in:

\[
x^2 - 3x + 2
\]

### Step 2: Factor the Numerator

Now, we will factor \( x^2 - 3x + 2 \):

\[
x^2 - 3x + 2 = (x - 1)(x - 2)
\]

### Step 3: Simplify the Denominator

Now, let's look at the denominator:

\[
(x - 2)(x - 1)(x + 2)(x - 2)(x + 1)(x + 2) = (x - 2)^2(x - 1)(x + 2)^2(x + 1)
\]

### Step 4: Rewrite the Complete Expression

Now substituting in the factored numerator gives:

\[
\frac{(x - 1)(x - 2)}{(x - 2)^2(x - 1)(x + 2)^2(x + 1)}
\]

### Step 5: Cancel Common Factors

In this expression, we can cancel out \( (x - 1) \) and one \( (x - 2) \):

\[
\frac{\cancel{(x - 1)} (x - 2)}{(x - 2) \cancel{(x - 1)} (x + 2)^2 (x + 1)} = \frac{1}{(x - 2)(x + 2)^2(x + 1)}
\]

### Final Result:

Thus, the expression that is equivalent to the given one is:

\[
\boxed{\frac{1}{(x - 2)(x + 2)^2(x + 1)}}
\]