Step 1: Simplify the Numerator

Question

First, let's focus on the top part, which is called the numerator:

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

We can combine like terms here. Group the \(x^2\) terms together and the \(x\) terms together:

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

Now, let's do the calculations:

- For \(3x^2 - 2x^2\), we get \(x^2\).
- For \(2x - 5x\), we get \(-3x\).

Putting it all together, we have:

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

### Step 2: Factor the Numerator

Next, we need to factor \(x^2 - 3x + 2\). We are looking for two numbers that multiply to \(2\) (the last number) and add up to \(-3\) (the middle number). Those numbers are \(-1\) and \(-2\). So, we can write:

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

### Step 3: Simplify the Denominator

Now, let's look at the bottom part, or the denominator:

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

Notice we can group the factors:

- We have two \( (x - 2) \) factors: \( (x - 2)(x - 2) = (x - 2)^2 \)
- We have two \( (x + 2) \) factors: \( (x + 2)(x + 2) = (x + 2)^2 \)

So, the denominator can be rewritten as:

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

### Step 4: Rewrite the Whole Expression

Now we can rewrite the whole expression using what we found:

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

### Step 5: Cancel Out Common Factors

In this fraction, we can cancel out what’s the same on the top and bottom:

1. The \( (x - 1) \) in the numerator and denominator cancels.
2. One \( (x - 2) \) in the numerator cancels with one \( (x - 2) \) in the denominator.

After canceling, we are left with:

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

### Final Answer

So, the final answer to our original expression is:

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

Answer 1

Certainly! Let’s streamline the explanation of the simplification process for the expression step-by-step:

Given Expression

Start with the expression:

\[ \frac{3x^2 + 2x - 5x - 2x^2 + 2}{(x - 2)(x - 1)(x + 2)(x - 2)(x + 1)(x + 2)} \]

Step 1: Simplify the Numerator

Combine like terms in the numerator:

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

Step 2: Factor the Numerator

Factor \(x^2 - 3x + 2\) to get:

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

Step 3: Simplify the Denominator

The denominator is:

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

Group factors:

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

Step 4: Rewrite the Whole Expression

Now the fraction is:

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

Step 5: Cancel Out Common Factors

Cancel \( (x - 1) \) and one \( (x - 2) \):

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

Final Answer

The final simplified expression is:

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