Binomial Theorem

To determine the absolute value of the difference between the coefficients of x4 and x6 in the expansion of the given expression, we start with:
x2−2x2+(x+1)4(x2−1)2
We simplify the function as follows:
f(x)=‌

2x2

(x2+1)(x2+2)

Expanding, we have:
f(x)=2x2(x2+1)−1(x2+2)−1
Simplifying further gives:
f(x)=‌

2x2

2

(1+x2)−1(1+‌

x2

2

)−1
This can be expanded into:
f(x)=x2[1−x2+x4−x6+⋯]⋅[1−‌

x2

2

+‌

x4

4

−‌

x4

8

+⋯]
Now, let's find the coefficients.
For x4 :
Coefficient calculation:
[−‌

1

2

−1]=‌

−3

2

For x6 :
Coefficient calculation:
[‌

1

4

+‌

1

2

+1]−‌

7

4

Now calculate the absolute difference:
‌‌ Difference ‌=|‌

7

4

−(‌

−3

2

)|=|‌

7

4

+‌

3

2

|
‌=|‌

7

4

+‌

6

4

|=|‌

13

4

|=‌

13

4

This process allows us to correctly determine the absolute value of the difference between the coefficients of x4 and x6 in the expansion.