To find the expression equivalent to \(\frac{3x^2 + 2x - 5x - 2x^2 + 2}{(x-2)(x-1)(x+2)(x-2)(x+1)(x+2)}\), we will first simplify the numerator.

Question

1. **Simplify the Numerator:**
\[
3x^2 + 2x - 5x - 2x^2 + 2
\]
Combine like terms:
\[
(3x^2 - 2x^2) + (2x - 5x) + 2 = x^2 - 3x + 2
\]

2. **Factor the Numerator:**
The expression \(x^2 - 3x + 2\) can be factored as follows:
\[
x^2 - 3x + 2 = (x-1)(x-2)
\]

3. **Identify and Simplify the Denominator:**
The denominator is:
\[
(x-2)(x-1)(x+2)(x-2)(x+1)(x+2)
\]
Notice that \((x-2)\) and \((x+2)\) repeat, so we can rewrite the denominator as:
\[
(x-1)(x-2)^2 (x+2)^2 (x+1)
\]

4. **Write the Complete Expression:**
Now we put the numerator and denominator together:
\[
\frac{(x-1)(x-2)}{(x-1)(x-2)^2 (x+2)^2 (x+1)}
\]

5. **Cancel Common Factors:**
We can cancel \((x-1)\) and \((x-2)\) from the numerator and denominator:
\[
\frac{1}{(x-2)(x+2)^2 (x+1)}
\]

Thus, the simplified expression is:
\[
\frac{1}{(x-2)(x+2)^2 (x+1)}
make this like a 7th grader wreote it

Answer 1

To simplify the expression

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

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!