Question

To simplify the expression
\[
\frac{3x^2 + 2x - 5x - 2x^2 + 2}{(x-2)(x-1)(x+2)(x-2)(x+1)(x+2)},
\]

let's break it down step by step!

### Step 1: Simplify the Numerator
We start with the numerator:

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

We can combine the like terms:
- The \(x^2\) terms: \(3x^2 - 2x^2 = 1x^2\) (or just \(x^2\))
- The \(x\) terms: \(2x - 5x = -3x\)
- And we have a constant term: \( +2\)

Putting all this together, we get:

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

### Step 2: Factor the Numerator
Now, we need to factor \(x^2 - 3x + 2\). We are looking for two numbers that multiply to \(2\) and add to \(-3\). These numbers are \(-1\) and \(-2\). So we can factor it as:

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

### Step 3: Simplify the Denominator
Next, let's look at the denominator:

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

Notice that \((x-2)\) appears twice and \((x+2)\) appears twice too. We can rewrite it as:

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

### Step 4: Put It All Together
Now we combine the simplified numerator and denominator:

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

### Step 5: Cancel Common Factors
Next, we can cancel out the common factors from the top and bottom. The \( (x-1) \) cancels out, and so does one \( (x-2) \):

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

### Final Answer
So, the final simplified expression is:

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

And that’s how you simplify that expression step-by-step!

make this smaller

Answer 1

To simplify the expression

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

we'll follow these steps:

### Step 1: Simplify the Numerator

Start with:

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

Combine like terms:

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

### Step 2: Factor the Numerator

Next, factor \(x^2 - 3x + 2\):

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

### Step 3: Simplify the Denominator

The denominator is:

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

This can be rewritten as:

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

### Step 4: Combine and Simplify

Now, we combine the simplified numerator and denominator:

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

### Step 5: Cancel Common Factors

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

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

### Final Answer

The simplified expression is:

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