Để giải các bài toán khai triển hằng đẳng thức này, chúng ta sẽ làm từng bài một:
1. \( (a-5b)^2 - 16b^2 \)
\[ (a-5b)^2 = a^2 - 10ab + 25b^2 \]
\[ (a-5b)^2 - 16b^2 = a^2 - 10ab + 25b^2 - 16b^2 \]
\[ (a-5b)^2 - 16b^2 = a^2 - 10ab + 9b^2 \]
2. \( 49a^2 - (2a-b)^2 \)
\[ (2a-b)^2 = 4a^2 - 4ab + b^2 \]
\[ 49a^2 - (2a-b)^2 = 49a^2 - (4a^2 - 4ab + b^2) \]
\[ 49a^2 - (2a-b)^2 = 49a^2 - 4a^2 + 4ab - b^2 \]
\[ 49a^2 - (2a-b)^2 = 45a^2 + 4ab - b^2 \]
3. \( (4a+3b)^2 - (b-2a)^2 \)
\[ (4a+3b)^2 = 16a^2 + 24ab + 9b^2 \]
\[ (b-2a)^2 = b^2 - 4ab + 4a^2 \]
\[ (4a+3b)^2 - (b-2a)^2 = 16a^2 + 24ab + 9b^2 - (b^2 - 4ab + 4a^2) \]
\[ (4a+3b)^2 - (b-2a)^2 = 16a^2 + 24ab + 9b^2 - b^2 + 4ab - 4a^2 \]
\[ (4a+3b)^2 - (b-2a)^2 = 12a^2 + 28ab + 8b^2 \]
4. \( 9(a+b)^2 - 4(a-2b)^2 \)
\[ (a+b)^2 = a^2 + 2ab + b^2 \]
\[ (a-2b)^2 = a^2 - 4ab + 4b^2 \]
\[ 9(a+b)^2 - 4(a-2b)^2 = 9(a^2 + 2ab + b^2) - 4(a^2 - 4ab + 4b^2) \]
\[ 9(a+b)^2 - 4(a-2b)^2 = 9a^2 + 18ab + 9b^2 - 4a^2 + 16ab - 16b^2 \]
\[ 9(a+b)^2 - 4(a-2b)^2 = 5a^2 + 34ab - 7b^2 \]
5. \( 4x^4 + 20x^2 + 25 \)
\[ 4x^4 + 20x^2 + 25 = (2x^2 + 5)^2 \]
6. \( 25x^2 - 20xy + 4y^2 \)
\[ 25x^2 - 20xy + 4y^2 = (5x - 2y)^2 \]
7. \( x^{10} - 4x^8 + 4x^6 \)
\[ x^{10} - 4x^8 + 4x^6 = x^6(x^4 - 4x^2 + 4) \]
\[ x^{10} - 4x^8 + 4x^6 = x^6(x^2 - 2)^2 \]
8. \( m^3 + 27 \)
\[ m^3 + 27 = (m + 3)(m^2 - 3m + 9) \]
9. \( 125 - (x+2)^3 \)
\[ 125 - (x+2)^3 = 125 - (x+2)^3 \]
10. \( (x-5)^3 - 27 \)
\[ (x-5)^3 - 27 = (x-5)^3 - 3^3 \]
\[ (x-5)^3 - 27 = (x-5 - 3)((x-5)^2 + (x-5) \cdot 3 + 3^2) \]
\[ (x-5)^3 - 27 = (x-8)(x^2 - 10x + 25) \]
11. \( x^{12} - y^4 \)
\[ x^{12} - y^4 = (x^6 - y^2)(x^6 + y^2) \]
12. \( x^3 + 3x^2 + 3x + 1 \)
\[ x^3 + 3x^2 + 3x + 1 = (x+1)^3 \]
13. \( \frac{1}{25} - 36x^2 \)
\[ \frac{1}{25} - 36x^2 = \frac{1 - 900x^2}{25} \]
14. \( (a-2b)^2 - 4b^2 \)
\[ (a-2b)^2 - 4b^2 = (a^2 - 4ab + 4b^2) - 4b^2 = a^2 - 4ab \]
15. \( 25a^2 - (a-b)^2 \)
\[ 25a^2 - (a-b)^2 = 25a^2 - (a^2 - 2ab + b^2) = 24a^2 + 2ab - b^2 \]
16. \( 36a^2 - (3a-2b)^2 \)
\[ 36a^2 - (3a-2b)^2 = 36a^2 - (9a^2 - 12ab + 4b^2) = 36a^2 - 9a^2 + 12ab - 4b^2 = 27a^2 + 12ab - 4b^2 \]
17. \( (5a-b)^2 - (2a+3b)^2 \)
\[ (5a-b)^2 - (2a+3b)^2 = (25a^2 - 10ab + b^2) - (4a^2 + 12ab + 9b^2) = 21a^2 - 22ab - 8b^2 \]
18. \( (2a-b)^2 - 4(a-b)^2 \)
\[ (2a-b)^2 - 4(a-b)^2 = (4a^2 - 4ab + b^2) - 4(a^2 - 2ab + b^2) = 4a^2 - 4ab + b^2 - 4a^2 + 8ab - 4b^2 = 4ab - 3b^2 \]
19. \( 49(x-5)^2 - (x+4)^2 \)
\[ 49(x-5)^2 - (x+4)^2 = 49(x^2 - 10x + 25) - (x^2 + 8x + 16) = 49x^2 - 490x + 1225 - x^2 - 8x - 16 = 48x^2 - 498x + 1209 \]
20. \( 36(x-t)^2 - 25(2x-1)^2 \)
\[ 36(x-t)^2 - 25(2x-1)^2 = 36(x^2 - 2xt + t^2) - 25(4x^2 - 4x + 1) = 36x^2 - 72xt + 36t^2 - 100x^2 + 100x - 25 \]
21. \( x^2 + 8x + 16 \)
\[ x^2 + 8x + 16 = (x + 4)^2 \]
22. \( 4x^2 + 12x + 9 \)
\[ 4x^2 + 12x + 9 = (2x + 3)^2 \]
23. \( 9x^4 + 24x^2 + 16 \)
\[ 9x^4 + 24x^2 + 16 = (3x^2 + 4)^2 \]
24. \( 4x^2 - 12xy + 9y^2 \)
\[ 4x^2 - 12xy + 9y^2 = (2x - 3y)^2 \]
25. \( 9x^4 - 12x^3 + 4x^2 \)
\[ 9x^4 - 12x^3 + 4x^2 = x^2(9x^2 - 12x + 4) \]
26. \( 8x^3 + 27y^3 \)
\[ 8x^3 + 27y^3 = (2x)^3 + (3y)^3 = (2x + 3y)(4x^2 - 6xy + 9y^2) \]
27. \( 8x^6 + 27y^3 \)
\[ 8x^6 + 27y^3 = (2x^2)^3 + (3y)^3 = (2x^2 + 3y)(4x^4 - 6x^2y + 9y^2) \]
28. \( \frac{1}{64}x^6 - 125y^3 \)
\[ \frac{1}{64}x^6 - 125y^3 = \frac{1}{64}x^6 - (5y)^3 = \left(\frac{1}{4}x^2 - 5y\right)\left(\frac{1}{16}x^4 + \frac{5}{4}x^2 + 25y^2\right) \]
29. \( x^3 - (y-1)^3 \)
\[ x^3 - (y-1)^3 = x^3 - (y^3 - 3y^2 + 3y - 1) = x^3 - y^3 + 3y^2 - 3y + 1 \]
30. \( (x+3)^3 - 125 \)
\[ (x+3)^3 - 125 = (x+3)^3 - 5^3 = (x+3 - 5)((x+3)^2 + (x+3) \cdot 5 + 5^2) \]
\[ (x+3)^3 - 125 = (x-2)(x^2 + 6x + 14) \]
31. \( x^6 - y^6 \)
\[ x^6 - y^6 = (x^3)^2 - (y^3)^2 = (x^3 - y^3)(x^3 + y^3) \]
32. \( x^3 + 6x^2 + 12x + 8 \)
\[ x^3 + 6x^2 + 12x + 8 = (x + 2)^3 \]
33. \( x^3 + 15x^2 + 75x + 125 \)
\[ x^3 + 15x^2 + 75x + 125 = (x + 5)^3 \]
34. \( m^3 - 6m^2 + 12m + 8 \)
\[ m^3 - 6m^2 + 12m + 8 = (m + 2)^3 \]
35. \( 8a^3 + 12a^2 + 1 \)
\[ 8a^3 + 12a^2 + 1 = (2a)^3 + (1)^3 = (2a + 1)(4a^2 + 4a + 1) \]
36. \( 64 - 48m + 12m^2 - m^3 \)
\[ 64 - 48m + 12m^2 - m^3 = (4 - m)^3 \]
37. \( 27a^3 - 27a^2 + 9a - 1 \)
\[ 27a^3 - 27a^2 + 9a - 1 = (3a - 1)^3 \]
38. \( 8a^3 - 12a^2 + 6a - 1 \)
\[ 8a^3 - 12a^2 + 6a - 1 = (2a - 1)^3 \]
39. \( 81a^2 - (5a-3b)^2 \)
\[ 81a^2 - (5a-3b)^2 = (9a - (5a - 3b))(9a + (5a - 3b)) \]
\[ 81a^2 - (5a-3b)^2 = (4a + 3b)(14a - 3b) \]
40. \( \frac{4}{9}a^3 - \frac{25}{4} \)
\[ \frac{4}{9}a^3 - \frac{25}{4} = \left(\frac{2}{3}a - \frac{5}{2}\right)\left(\frac{2}{3}a^2 + \frac{5}{3}a + \frac{25}{4}\right) \]
