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.