1. **State the problem:** We need to find the numerator of the expression using the FOIL method. The numerator is: $$(x^2 - 2x + 3)(\Delta x^2 + 2x \Delta x - 2 \Delta x)(x^2 + 2x - 3)(x^2 + 2x - 3)(\Delta x^2 + 2x \Delta x + 2 \Delta x)$$ 2. **Recall the FOIL method:** FOIL stands for First, Outer, Inner, Last and is used to multiply two binomials: $$ (a + b)(c + d) = ac + ad + bc + bd $$ 3. **Step-by-step multiplication:** We will multiply the first two factors first: $$(x^2 - 2x + 3)(\Delta x^2 + 2x \Delta x - 2 \Delta x)$$ Since the second factor has three terms, we distribute each term: $$x^2 \cdot \Delta x^2 + x^2 \cdot 2x \Delta x - x^2 \cdot 2 \Delta x - 2x \cdot \Delta x^2 - 2x \cdot 2x \Delta x + 2x \cdot 2 \Delta x + 3 \cdot \Delta x^2 + 3 \cdot 2x \Delta x - 3 \cdot 2 \Delta x$$ 4. **Simplify each term:** $$x^2 \Delta x^2 + 2x^3 \Delta x - 2x^2 \Delta x - 2x \Delta x^2 - 4x^2 \Delta x + 6x \Delta x + 3 \Delta x^2 + 6x \Delta x - 6 \Delta x$$ 5. **Combine like terms:** Group terms with $\Delta x^2$: $$x^2 \Delta x^2 - 2x \Delta x^2 + 3 \Delta x^2 = (x^2 - 2x + 3) \Delta x^2$$ Group terms with $\Delta x$: $$2x^3 \Delta x - 2x^2 \Delta x - 4x^2 \Delta x + 6x \Delta x + 6x \Delta x - 6 \Delta x = (2x^3 - 6x^2 + 12x - 6) \Delta x$$ 6. **Rewrite the result:** $$ (x^2 - 2x + 3) \Delta x^2 + (2x^3 - 6x^2 + 12x - 6) \Delta x $$ 7. **Next multiply by the third factor $(x^2 + 2x - 3)$:** Distribute each term: $$[(x^2 - 2x + 3) \Delta x^2] (x^2 + 2x - 3) + [(2x^3 - 6x^2 + 12x - 6) \Delta x] (x^2 + 2x - 3)$$ 8. **Multiply the first part:** $$(x^2 - 2x + 3)(x^2 + 2x - 3) \Delta x^2$$ Use FOIL on $(x^2 - 2x + 3)(x^2 + 2x - 3)$: $$x^2 \cdot x^2 + x^2 \cdot 2x - x^2 \cdot 3 - 2x \cdot x^2 - 2x \cdot 2x + 2x \cdot 3 + 3 \cdot x^2 + 3 \cdot 2x - 3 \cdot 3$$ Simplify: $$x^4 + 2x^3 - 3x^2 - 2x^3 - 4x^2 + 6x + 3x^2 + 6x - 9$$ Combine like terms: $$x^4 + (2x^3 - 2x^3) + (-3x^2 - 4x^2 + 3x^2) + (6x + 6x) - 9 = x^4 - 4x^2 + 12x - 9$$ So the first part is: $$(x^4 - 4x^2 + 12x - 9) \Delta x^2$$ 9. **Multiply the second part:** $$(2x^3 - 6x^2 + 12x - 6)(x^2 + 2x - 3) \Delta x$$ Multiply each term: $$2x^3 \cdot x^2 + 2x^3 \cdot 2x - 2x^3 \cdot 3 - 6x^2 \cdot x^2 - 6x^2 \cdot 2x + 6x^2 \cdot 3 + 12x \cdot x^2 + 12x \cdot 2x - 12x \cdot 3 - 6 \cdot x^2 - 6 \cdot 2x + 6 \cdot 3$$ Simplify: $$2x^5 + 4x^4 - 6x^3 - 6x^4 - 12x^3 + 18x^2 + 12x^3 + 24x^2 - 36x - 6x^2 - 12x + 18$$ Combine like terms: $$2x^5 + (4x^4 - 6x^4) + (-6x^3 - 12x^3 + 12x^3) + (18x^2 + 24x^2 - 6x^2) + (-36x - 12x) + 18$$ $$= 2x^5 - 2x^4 - 6x^3 + 36x^2 - 48x + 18$$ So the second part is: $$(2x^5 - 2x^4 - 6x^3 + 36x^2 - 48x + 18) \Delta x$$ 10. **Final numerator expression:** $$ (x^4 - 4x^2 + 12x - 9) \Delta x^2 + (2x^5 - 2x^4 - 6x^3 + 36x^2 - 48x + 18) \Delta x $$ **Answer:** The numerator expanded using FOIL and distribution is: $$ (x^4 - 4x^2 + 12x - 9) \Delta x^2 + (2x^5 - 2x^4 - 6x^3 + 36x^2 - 48x + 18) \Delta x $$