1. **State the problem:** Simplify the expression $$(5a^2 - 6 + 9)(2a - 3) - (2a^2 - 5a + 4)(5a + 1).$$ 2. **Simplify inside parentheses:** $$5a^2 - 6 + 9 = 5a^2 + 3.$$ So the expression becomes: $$(5a^2 + 3)(2a - 3) - (2a^2 - 5a + 4)(5a + 1).$$ 3. **Multiply the first pair:** $$(5a^2 + 3)(2a - 3) = 5a^2 \times 2a + 5a^2 \times (-3) + 3 \times 2a + 3 \times (-3)$$ $$= 10a^3 - 15a^2 + 6a - 9.$$ 4. **Multiply the second pair:** $$(2a^2 - 5a + 4)(5a + 1) = 2a^2 \times 5a + 2a^2 \times 1 - 5a \times 5a - 5a \times 1 + 4 \times 5a + 4 \times 1$$ $$= 10a^3 + 2a^2 - 25a^2 - 5a + 20a + 4$$ $$= 10a^3 - 23a^2 + 15a + 4.$$ 5. **Subtract the second product from the first:** $$ (10a^3 - 15a^2 + 6a - 9) - (10a^3 - 23a^2 + 15a + 4)$$ $$= 10a^3 - 15a^2 + 6a - 9 - 10a^3 + 23a^2 - 15a - 4.$$ 6. **Combine like terms:** $$ (10a^3 - \cancel{10a^3}) + (-15a^2 + 23a^2) + (6a - 15a) + (-9 - 4)$$ $$= 0 + 8a^2 - 9a - 13.$$ 7. **Final simplified expression:** $$\boxed{8a^2 - 9a - 13}.$$