Find the coefficient of `x^7` in `(1+ x^2)^4 (1 + x)^7`

Answer

To find the coefficient of \( x^7 \) in the expression \( (1 + x^2)^4 (1 + x)^7 \), we can use the binomial theorem to expand both parts of the expression and then combine the results.

### Step-by-Step Solution:

1. **Expand \( (1 + x^2)^4 \)**:
Using the binomial theorem, we have:
\[ (1 + x^2)^4 = \sum_{k=0}^{4} \binom{4}{k} (x^2)^k (1)^{4-k} = \sum_{k=0}^{4} \binom{4}{k} x^{2k} \]
This gives us the terms:
- For \( k = 0 \): \( \binom{4}{0} x^0 = 1 \)
- For \( k = 1 \): \( \binom{4}{1} x^2 = 4x^2 \)
- For \( k = 2 \): \( \binom{4}{2} x^4 = 6x^4 \)
- For \( k = 3 \): \( \binom{4}{3} x^6 = 4x^6 \)
- For \( k = 4 \): \( \binom{4}{4} x^8 = x^8 \)

So, we have:
\[ (1 + x^2)^4 = 1 + 4x^2 + 6x^4 + 4x^6 + x^8 \]

2. **Expand \( (1 + x)^7 \)**:
Again, using the binomial theorem:
\[ (1 + x)^7 = \sum_{j=0}^{7} \binom{7}{j} x^j \]
This gives us the terms:
- For \( j = 0 \): \( \binom{7}{0} x^0 = 1 \)
- For \( j = 1 \): \( \binom{7}{1} x^1 = 7x \)
- For \( j = 2 \): \( \binom{7}{2} x^2 = 21x^2 \)
- For \( j = 3 \): \( \binom{7}{3} x^3 = 35x^3 \)
- For \( j = 4 \): \( \binom{7}{4} x^4 = 35x^4 \)
- For \( j = 5 \): \( \binom{7}{5} x^5 = 21x^5 \)
- For \( j = 6 \): \( \binom{7}{6} x^6 = 7x^6 \)
- For \( j = 7 \): \( \binom{7}{7} x^7 = 1x^7 \)

So, we have:
\[ (1 + x)^7 = 1 + 7x + 21x^2 + 35x^3 + 35x^4 + 21x^5 + 7x^6 + x^7 \]

3. **Combine the two expansions**:
We need to find the coefficient of \( x^7 \) in the product:
\[ (1 + 4x^2 + 6x^4 + 4x^6 + x^8)(1 + 7x + 21x^2 + 35x^3 + 35x^4 + 21x^5 + 7x^6 + x^7) \]

We can find the combinations of terms from each expansion that will give us \( x^7 \):
- From \( 1 \) (from \( (1 + x^2)^4 \)) and \( x^7 \) (from \( (1 + x)^7 \)): Coefficient = \( 1 \cdot 1 = 1 \)
- From \( 4x^2 \) (from \( (1 + x^2)^4 \)) and \( 21x^5 \) (from \( (1 + x)^7 \)): Coefficient = \( 4 \cdot 21 = 84 \)
- From \( 6x^4 \) (from \( (1 + x^2)^4 \)) and \( 35x^3 \) (from \( (1 + x)^7 \)): Coefficient = \( 6 \cdot 35 = 210 \)
- From \( 4x^6 \) (from \( (1 + x^2)^4 \)) and \( 7x \) (from \( (1 + x)^7 \)): Coefficient = \( 4 \cdot 7 = 28 \)

4. **Sum the coefficients**:
Now we add all the coefficients we found:
\[ 1 + 84 + 210 + 28 = 323 \]

### Final Answer:
The coefficient of \( x^7 \) in \( (1 + x^2)^4 (1 + x)^7 \) is \( \boxed{323} \).