127. Представьте в виде многочлена выражение: 1) (a + 2)²; 2) (6 – x)²; 3) (1/2 a + b)²; 4) (3x - 4)²; 5) (5m + 3n)²; 6) (0,1a + 10b)²; 7) (6x - 1/3 y)²; 8) (n² + 1)²; 9) (x⁴ - x²)²; 10) (y⁴ + y³)²; 11) (-3a + 4b³)²; 12) (-2 - 5x)²; 13) (1/3 m + 3/5 n)²; 14) (6ab² - a²b)²; 15) (5a⁴ - 2a²b⁴)².

Решение:

1. \( (a + 2)^2 = a^2 + 2 \cdot a \cdot 2 + 2^2 = a^2 + 4a + 4 \)
2. \( (6 - x)^2 = 6^2 - 2 \cdot 6 \cdot x + x^2 = 36 - 12x + x^2 \)
3. \( \left(\frac{1}{2} a + b\right)^2 = \left(\frac{1}{2} a\right)^2 + 2 \cdot \frac{1}{2} a \cdot b + b^2 = \frac{1}{4} a^2 + ab + b^2 \)
4. \( (3x - 4)^2 = (3x)^2 - 2 \cdot 3x \cdot 4 + 4^2 = 9x^2 - 24x + 16 \)
5. \( (5m + 3n)^2 = (5m)^2 + 2 \cdot 5m \cdot 3n + (3n)^2 = 25m^2 + 30mn + 9n^2 \)
6. \( (0.1a + 10b)^2 = (0.1a)^2 + 2 \cdot 0.1a \cdot 10b + (10b)^2 = 0.01a^2 + 2ab + 100b^2 \)
7. \( \left(6x - \frac{1}{3} y\right)^2 = (6x)^2 - 2 \cdot 6x \cdot \frac{1}{3} y + \left(\frac{1}{3} y\right)^2 = 36x^2 - 4xy + \frac{1}{9} y^2 \)
8. \( (n^2 + 1)^2 = (n^2)^2 + 2 \cdot n^2 \cdot 1 + 1^2 = n^4 + 2n^2 + 1 \)
9. \( (x^4 - x^2)^2 = (x^4)^2 - 2 \cdot x^4 \cdot x^2 + (x^2)^2 = x^8 - 2x^6 + x^4 \)
10. \( (y^4 + y^3)^2 = (y^4)^2 + 2 \cdot y^4 \cdot y^3 + (y^3)^2 = y^8 + 2y^7 + y^6 \)
11. \( (-3a + 4b^3)^2 = (-3a)^2 + 2 \cdot (-3a) \cdot 4b^3 + (4b^3)^2 = 9a^2 - 24ab^3 + 16b^6 \)
12. \( (-2 - 5x)^2 = (-2)^2 - 2 \cdot (-2) \cdot 5x + (5x)^2 = 4 + 20x + 25x^2 \)
13. \( \left(\frac{1}{3} m + \frac{3}{5} n\right)^2 = \left(\frac{1}{3} m\right)^2 + 2 \cdot \frac{1}{3} m \cdot \frac{3}{5} n + \left(\frac{3}{5} n\right)^2 = \frac{1}{9} m^2 + \frac{2}{5} mn + \frac{9}{25} n^2 \)
14. \( (6ab^2 - a^2b)^2 = (6ab^2)^2 - 2 \cdot 6ab^2 \cdot a^2b + (a^2b)^2 = 36a^2b^4 - 12a^3b^3 + a^4b^2 \)
15. \( (5a^4 - 2a^2b^4)^2 = (5a^4)^2 - 2 \cdot 5a^4 \cdot 2a^2b^4 + (2a^2b^4)^2 = 25a^8 - 20a^6b^4 + 4a^4b^8 \)

Ответ: 1) \( a^2 + 4a + 4 \); 2) \( 36 - 12x + x^2 \); 3) \( \frac{1}{4} a^2 + ab + b^2 \); 4) \( 9x^2 - 24x + 16 \); 5) \( 25m^2 + 30mn + 9n^2 \); 6) \( 0.01a^2 + 2ab + 100b^2 \); 7) \( 36x^2 - 4xy + \frac{1}{9} y^2 \); 8) \( n^4 + 2n^2 + 1 \); 9) \( x^8 - 2x^6 + x^4 \); 10) \( y^8 + 2y^7 + y^6 \); 11) \( 9a^2 - 24ab^3 + 16b^6 \); 12) \( 4 + 20x + 25x^2 \); 13) \( \frac{1}{9} m^2 + \frac{2}{5} mn + \frac{9}{25} n^2 \); 14) \( 36a^2b^4 - 12a^3b^3 + a^4b^2 \); 15) \( 25a^8 - 20a^6b^4 + 4a^4b^8 \).