1. **State the problem:** We want to determine if the expression $$\frac{(x^3 - 5x)^2 + 7x - 2}{x - 2}$$ can be factored and simplified by canceling out common factors. 2. **Rewrite the numerator:** The numerator is $$((x^3 - 5x)^2 + 7x - 2)$$. First, expand or simplify the numerator if possible. 3. **Expand $(x^3 - 5x)^2$:** $$ (x^3 - 5x)^2 = (x^3)^2 - 2 \cdot x^3 \cdot 5x + (5x)^2 = x^6 - 10x^4 + 25x^2 $$ 4. **Rewrite numerator fully:** $$ x^6 - 10x^4 + 25x^2 + 7x - 2 $$ 5. **Check if numerator is divisible by $(x-2)$:** Use polynomial division or the Remainder Theorem. 6. **Remainder Theorem:** Substitute $x=2$ into numerator: $$ 2^6 - 10 \cdot 2^4 + 25 \cdot 2^2 + 7 \cdot 2 - 2 = 64 - 160 + 100 + 14 - 2 = 16 $$ Since remainder is 16 (not zero), numerator is not divisible by $(x-2)$. 7. **Conclusion:** Since numerator is not divisible by denominator, the expression cannot be factored to cancel out $(x-2)$. **Final answer:** The expression $$\frac{(x^3 - 5x)^2 + 7x - 2}{x - 2}$$ cannot be simplified by factoring and canceling out $(x-2)$.