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.

Question

### 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
make this like a 7th grader wrote 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, just like you'd do in seventh grade math!

Step 1: Simplify the Numerator

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:

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

Yay! We simplified it!