Final answer:
The 2nd term in the expansion of (a + b)^4(a - b)^4 is found using the binomial theorem and squaring methods, resulting in the term a^2b^2. The correct option is A).
Step-by-step explanation:
To find the second term in the expansion of (a + b)^4(a - b)^4, we must first realize that this is equivalent to raising the binomial (a + b)(a - b) to the fourth power, since the multiplication of two binomials can be handled this way. Squaring (a + b)(a - b) gives us (a^2 - b^2), and therefore, the expression becomes (a^2 - b^2)^4.
To get to the second term of this expanded expression, we use the binomial theorem. The second term of the binomial expansion is given by 4C1(a^2)^(4-1)(-b^2)^1 where 4C1 is the combination of 4 items taken 1 at a time.
Calculating this gives us 4 * (a^8 / a^2) * (-b^2) = 4 * a^6 * (-b^2). The negative sign and one of the b^2 cancel out, resulting in -4a^6b^2, but since we are looking for the second term of (a + b)^4(a - b)^4, we then square a^6b^2 to get the second term (a^6b^2)^2 = a^12b^4.