Step 1: Understanding the Concept:
For a binomial expansion \( (1+x)^n \), if \( n \) is even, the middle term is \( T_{\frac{n}{2} + 1} \).
Step 2: Key Formula or Approach:
1. Middle term of \( (1 + \alpha x)^4 \): \( T_{\frac{4}{2} + 1} = T_3 = \binom{4}{2}(\alpha x)^2 \).
2. Middle term of \( (1 - \alpha x)^6 \): \( T_{\frac{6}{2} + 1} = T_4 = \binom{6}{3}(-\alpha x)^3 \).
Step 3: Detailed Explanation:
Coefficient of middle term of \( (1 + \alpha x)^4 \):
\[ C_1 = \binom{4}{2}\alpha^2 = \frac{4 \times 3}{2} \alpha^2 = 6\alpha^2 \]
Coefficient of middle term of \( (1 - \alpha x)^6 \):
\[ C_2 = \binom{6}{3}(-\alpha)^3 = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} (-\alpha^3) = -20\alpha^3 \]
According to the question, \( C_1 = C_2 \):
\[ 6\alpha^2 = -20\alpha^3 \]
Assuming \( \alpha \neq 0 \):
\[ 6 = -20\alpha \]
\[ \alpha = -\frac{6}{20} = -\frac{3}{10} \]
Step 4: Final Answer:
The value of \( \alpha \) is \(-3/10\).