1. **Problem K:** Simplify and solve $$5x^2 \cdot (3x^2 + 1)^4 \cdot 6x + (3x^2 + 1)^5 \cdot 2x$$. 2. **Step 1:** Write the expression clearly: $$5x^2 \cdot (3x^2 + 1)^4 \cdot 6x + (3x^2 + 1)^5 \cdot 2x$$ 3. **Step 2:** Multiply the terms in the first product: $$5x^2 \cdot 6x = 30x^3$$ So the expression becomes: $$30x^3 \cdot (3x^2 + 1)^4 + 2x \cdot (3x^2 + 1)^5$$ 4. **Step 3:** Factor out the common term: $$x \cdot (3x^2 + 1)^4$$ from both terms: $$x \cdot (3x^2 + 1)^4 \left(30x^2 + 2(3x^2 + 1)\right)$$ 5. **Step 4:** Simplify inside the parentheses: $$30x^2 + 2(3x^2 + 1) = 30x^2 + 6x^2 + 2 = 36x^2 + 2$$ 6. **Step 5:** Final simplified expression: $$x \cdot (3x^2 + 1)^4 \cdot (36x^2 + 2)$$ --- 7. **Problem J:** Simplify and solve $$2 \cdot (x - 1) \cdot (2x + 2)^3 \cdot [4(x - 1) + (2x + 2)]$$. 8. **Step 1:** Write the expression clearly: $$2 \cdot (x - 1) \cdot (2x + 2)^3 \cdot [4(x - 1) + (2x + 2)]$$ 9. **Step 2:** Simplify inside the bracket: $$4(x - 1) + (2x + 2) = 4x - 4 + 2x + 2 = 6x - 2$$ 10. **Step 3:** Substitute back: $$2 \cdot (x - 1) \cdot (2x + 2)^3 \cdot (6x - 2)$$ 11. **Step 4:** Factor out 2 from $(2x + 2)$: $$2x + 2 = 2(x + 1)$$ So, $$(2x + 2)^3 = (2(x + 1))^3 = 2^3 (x + 1)^3 = 8 (x + 1)^3$$ 12. **Step 5:** Substitute and multiply constants: $$2 \cdot (x - 1) \cdot 8 (x + 1)^3 \cdot (6x - 2) = 16 (x - 1) (x + 1)^3 (6x - 2)$$ 13. **Step 6:** Factor 2 from $(6x - 2)$: $$6x - 2 = 2(3x - 1)$$ 14. **Step 7:** Final expression: $$16 (x - 1) (x + 1)^3 \cdot 2 (3x - 1) = 32 (x - 1) (x + 1)^3 (3x - 1)$$