find the coefficient of x^2 in the expansion of (1+x^2)(x/2 - 4/x)^6?

Method 1:

(1 + x²)[(x/2) - (4/x)]⁶
= (1 + x²)[(x/2)⁶ - ₆C₁ (x/2)⁵ (4/x) + ₆C₂ (x/2)⁴ (4/x)² - ₆C₃ (x/2)³ (4/x)³ + ₆C₄ (x/2)² (4/x)⁴ - ₆C₅ (x/2) (4/x)⁵ + (4/x)⁶]
= (1 + x²)[…… + ₆C₂ (x/2)⁴ (4/x)² - ₆C₃ (x/2)³ (4/x)³ + ……]
= (1 + x²)[…… + 15 (x⁴/16) (16/x²) - 20 (x³/8) (64/x³) + ……]
= (1 + x²)[…… + 15x² - 160 + ……]

The coefficient of x² in the expansion of (1 + x²)[(x/2) - (4/x)]⁶
= 15 - 160
= -145

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Method 2:

The nth term of [(x/2) - (4/x)]⁶ = (-1)ⁿ⁺¹ ₆Cₙ₋₁ (x/2)⁶⁻ⁿ⁺¹ (4/x)ⁿ⁻¹

When [(x/2) - (4/x)]⁶ = x⁰
6 - n + 1 - (n - 1) = 0
n = 4
The constant term of [(x/2) - (4/x)]⁶ = (-1)⁵ ₆C₃ (x/2)³ (4/x)³ = -160

When [(x/2) - (4/x)]⁶x⁶⁻ⁿ⁺¹ (1/x)ⁿ⁻¹ = x²
6 - n + 1 - (n - 1) = 2
n = 3
The x² term of [(x/2) - (4/x)]⁶= (-1)⁴ ₆C₂ (x/2)⁴ (4/x)² = 15

(1 + x²)[(x/2) - (4/x)]⁶
= (1 + x²)(-160 + 15x² + …… )

The coefficient of x² in the expansion of (1 + x²)[(x/2) - (4/x)]⁶
= 15 - 160
= -145

-145 is the coefficient of x^2 in the expansion of (1 + x^2)(x/2 - 4/x)^6

(x/2 - 4/x)^6 =>
((x^2 - 8) / (2x))^6 =>
(x^2 - 8)^6 / (64 * x^6)

1 * (x^2 - 8)^6 / (64 * x^6)

and

x^2 * (x^2 - 8)^6 / (64 * x^6) =>
(x^2 - 8)^6 / (64 * x^4)

We need to know when we have x^6 and when we have x^8

(a - b)^n = nC0 * a^n * (b)^0 + nC1 * a^(n - 1) * (b)^1 + .... + nCn * a^(n - n) * (b)^n

(x^2 - 8)^6

a = x^2
b = -8
n = 6

6Ck * (x^2)^(6 - k) * (-8)^k =>
(6! / (k! * (6 - k)!)) * x^(12 - 2k) * (-8)^k

x^(12 - 2k) = x^8 , x^6
12 - 2k = 8 , 6
4 , 6 = 2k
2 , 3 = k

6C3 * (x^2)^(6 - 3) * (-8)^3 =>
(6! / (3! * 3!)) * x^(6) * (-512) =>
20 * (-512) * x^6

6C2 * (x^2)^(6 - 2) * (-8)^2 =>
(6! / (4! * 2!)) * x^8 * 64 =>
15 * 64 * x^8

Now we have:

1 * x^8 * 15 * 64 / (64 * x^6) + x^6 * 20 * (-512) / (64 * x^4) =>
15 * x^2 - 20 * 8 * x^2 =>
15 * x^2 - 160 * x^2 =>
-145 * x^2

https://www.wolframalpha.com/input/?i=%281+%2B+x%5E2%29+*+%28%28x%2F2%29+-+%284%2Fx%29%29%5E6

The coefficient is -145