1. **Stating the problem:** Simplify the expression $$(x^2 - 2x + 3)(\Delta x^2 + 2x \Delta x - 2 \Delta x)(x^2 + 2x - 3)(x^2 + 2x - 3)(\Delta x^2 + 2x \Delta x + 2 \Delta x) \frac{x^2 + 2x - 3}{\Delta x (\Delta x^2 + 2x \Delta x + 2 \Delta x + x^2 + 2x - 3)(x^2 - 2x + 3)}$$ 2. **Understanding the notation:** Here, $\Delta x$ likely represents a small increment or difference in $x$. We treat it as a variable. 3. **Rewrite the expression clearly:** $$E = (x^2 - 2x + 3)(\Delta x^2 + 2x \Delta x - 2 \Delta x)(x^2 + 2x - 3)^2 (\Delta x^2 + 2x \Delta x + 2 \Delta x) \times \frac{x^2 + 2x - 3}{\Delta x (\Delta x^2 + 2x \Delta x + 2 \Delta x + x^2 + 2x - 3)(x^2 - 2x + 3)}$$ 4. **Factor where possible:** - Factor $x^2 + 2x - 3$: $$x^2 + 2x - 3 = (x + 3)(x - 1)$$ - Factor $x^2 - 2x + 3$ cannot be factored nicely over reals. 5. **Cancel common factors:** - $(x^2 - 2x + 3)$ appears in numerator and denominator, cancel: $$\cancel{(x^2 - 2x + 3)} \cdots / \cancel{(x^2 - 2x + 3)}$$ - $(x^2 + 2x - 3)$ appears squared in numerator and once in denominator, cancel one: $$\frac{(x^2 + 2x - 3)^2}{(x^2 + 2x - 3)} = x^2 + 2x - 3$$ 6. **Rewrite $\Delta x^2 + 2x \Delta x + 2 \Delta x + x^2 + 2x - 3$ as sum:** Group terms: $$\Delta x^2 + 2x \Delta x + 2 \Delta x + x^2 + 2x - 3 = (x^2 + 2x - 3) + (\Delta x^2 + 2x \Delta x + 2 \Delta x)$$ 7. **Substitute back and simplify:** Expression becomes: $$E = (\Delta x^2 + 2x \Delta x - 2 \Delta x)(x^2 + 2x - 3)(\Delta x^2 + 2x \Delta x + 2 \Delta x) \times \frac{1}{\Delta x ((x^2 + 2x - 3) + (\Delta x^2 + 2x \Delta x + 2 \Delta x))}$$ 8. **Factor $\Delta x$ from terms with $\Delta x$:** - $\Delta x^2 + 2x \Delta x - 2 \Delta x = \Delta x(\Delta x + 2x - 2)$ - $\Delta x^2 + 2x \Delta x + 2 \Delta x = \Delta x(\Delta x + 2x + 2)$ - Denominator term $\Delta x^2 + 2x \Delta x + 2 \Delta x + x^2 + 2x - 3 = (x^2 + 2x - 3) + \Delta x(\Delta x + 2x + 2)$ 9. **Rewrite $E$ with factored $\Delta x$:** $$E = \frac{\Delta x(\Delta x + 2x - 2)(x^2 + 2x - 3) \Delta x(\Delta x + 2x + 2)}{\Delta x ((x^2 + 2x - 3) + \Delta x(\Delta x + 2x + 2))}$$ 10. **Cancel one $\Delta x$ from numerator and denominator:** $$E = \frac{\cancel{\Delta x}(\Delta x + 2x - 2)(x^2 + 2x - 3) \Delta x(\Delta x + 2x + 2)}{\cancel{\Delta x} ((x^2 + 2x - 3) + \Delta x(\Delta x + 2x + 2))}$$ 11. **Final simplified form:** $$E = \frac{(\Delta x + 2x - 2)(x^2 + 2x - 3) \Delta x (\Delta x + 2x + 2)}{(x^2 + 2x - 3) + \Delta x (\Delta x + 2x + 2)}$$ This is the simplified expression. --- **Final answer:** $$\boxed{E = \frac{(\Delta x + 2x - 2)(x^2 + 2x - 3) \Delta x (\Delta x + 2x + 2)}{(x^2 + 2x - 3) + \Delta x (\Delta x + 2x + 2)}}$$