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