1. We start with the expression $(4x^2 - 16x + 7)(4x^2 - 16x + 15) + 16$. 2. Multiply the two quadratics: $$ (4x^2 - 16x + 7)(4x^2 - 16x + 15) = 16x^4 - 128x^3 + 148x^2 - 208x + 105. $$ 3. Add 16 to the product: $$ 16x^4 - 128x^3 + 148x^2 - 208x + 105 + 16 = 16x^4 - 128x^3 + 148x^2 - 208x + 121. $$ --- 4. Next, consider $(9x^2 + 9x - 4)(9x^2 + 9x - 10) - 72$. 5. Multiply: $$ (9x^2 + 9x - 4)(9x^2 + 9x - 10) = 81x^4 + 162x^3 - 153x^2 - 54x + 40. $$ 6. Subtract 72: $$ 81x^4 + 162x^3 - 153x^2 - 54x + 40 - 72 = 81x^4 + 162x^3 - 153x^2 - 54x - 32. $$ --- 7. Simplify $(x + 2)(x + 4)(x + 6)(x + 8) - 9$. 8. Multiply in pairs: $$ (x+2)(x+8) = x^2 + 10x +16, $$ $$ (x+4)(x+6) = x^2 + 10x + 24. $$ 9. Multiply the two quadratics: $$ (x^2 + 10x +16)(x^2 + 10x + 24) = x^4 + 20x^3 + 148x^2 + 400x + 384. $$ 10. Subtract 9: $$ x^4 + 20x^3 + 148x^2 + 400x + 375. $$ --- 11. Simplify $x(x + 1)(x + 2)(x + 3) + 1$. 12. Multiply $(x+1)(x+2) = x^2 + 3x + 2$. 13. Multiply $(x^2 + 3x + 2)(x + 3) = x^3 + 6x^2 + 11x + 6$. 14. Multiply by $x$: $$ x(x^3 + 6x^2 + 11x + 6) = x^4 + 6x^3 + 11x^2 + 6x. $$ 15. Add 1: $$ x^4 + 6x^3 + 11x^2 + 6x + 1. $$ --- 16. Simplify $(x + 1)(x + 2)(x + 3)(x + 6) - 3x^2$. 17. Multiply $(x+1)(x+6) = x^2 + 7x + 6$. 18. Multiply $(x+2)(x+3) = x^2 + 5x + 6$. 19. Multiply these quadratics: $$ (x^2 + 7x + 6)(x^2 + 5x + 6) = x^4 + 12x^3 + 47x^2 + 72x + 36. $$ 20. Subtract $3x^2$: $$ x^4 + 12x^3 + 44x^2 + 72x + 36. $$ --- 21. For $64x^3 - 144x^2y + 108xy^2 - 27y^3$, recognize it as a cubic expression. 22. Notice it matches the formula for $(4x - 3y)^3$ since: $$ (4x)^3 = 64x^3, $$ $$ 3(4x)^2(-3y) = -144x^2y, $$ $$ 3(4x)(-3y)^2 = 108xy^2, $$ $$ (-3y)^3 = -27y^3. $$ 23. So, $$ 64x^3 - 144x^2y + 108xy^2 - 27y^3 = (4x - 3y)^3. $$