The coefficient of `x^5` in the expansion of `(1+x^2)(1+x)^4` is (a) 12 (b) 5 (c) 4 (d) 56

Options

A. `12`
B. `5`
C. `4`
D. `56`

Explanation

To find the coefficient of \( x^5 \) in the expansion of \( (1 + x^2)(1 + x)^4 \), we can follow these steps:

### Step 1: Expand \( (1 + x)^4 \) using the Binomial Theorem
The Binomial Theorem states that:
\[ (1 + x)^n = \sum_{k=0}^{n} \binom{n}{k} x^k \]
For \( n = 4 \):
\[ (1 + x)^4 = \binom{4}{0} x^0 + \binom{4}{1} x^1 + \binom{4}{2} x^2 + \binom{4}{3} x^3 + \binom{4}{4} x^4 \]
Calculating the coefficients:
\[ = 1 + 4x + 6x^2 + 4x^3 + x^4 \]

### Step 2: Multiply by \( (1 + x^2) \)
Now we need to multiply this expansion by \( (1 + x^2) \):
\[ (1 + x^2)(1 + 4x + 6x^2 + 4x^3 + x^4) \]
Distributing \( (1 + x^2) \):
\[ = 1(1 + 4x + 6x^2 + 4x^3 + x^4) + x^2(1 + 4x + 6x^2 + 4x^3 + x^4) \]
This gives us:
\[ = 1 + 4x + 6x^2 + 4x^3 + x^4 + x^2 + 4x^3 + 6x^4 + 4x^5 + x^6 \]

### Step 3: Combine like terms
Now, we combine the like terms:
\[ = 1 + 4x + (6x^2 + x^2) + (4x^3 + 4x^3) + (x^4 + 6x^4) + 4x^5 + x^6 \]
This simplifies to:
\[ = 1 + 4x + 7x^2 + 8x^3 + 7x^4 + 4x^5 + x^6 \]

### Step 4: Identify the coefficient of \( x^5 \)
From the combined expression, we can see that the coefficient of \( x^5 \) is \( 4 \).

### Final Answer
Thus, the coefficient of \( x^5 \) in the expansion of \( (1 + x^2)(1 + x)^4 \) is \( \boxed{4} \).

---