(3 +x2)power n+1 when n is an interager given that the coefficint of xpower 4=20412 find n

The expression you're working with is \((3 + x^2)^{n+1}\), and you're tasked with finding \(n\), given that the coefficient of \(x^4\) is 20412.

### Step-by-Step Solution:

1. **General Binomial Expansion**:
   For an expression of the form \((a + b)^m\), the binomial expansion is given by:

   \[
   (a + b)^m = \sum_{k=0}^{m} \binom{m}{k} a^{m-k} b^k
   \]
   
   In your case, \(a = 3\), \(b = x^2\), and \(m = n+1\). So, the expansion of \((3 + x^2)^{n+1}\) would be:

   \[
   (3 + x^2)^{n+1} = \sum_{k=0}^{n+1} \binom{n+1}{k} 3^{n+1-k} (x^2)^k
   \]

2. **Identifying the \(x^4\) Term**:
   In the expansion, each term involving \(x\) is derived from \((x^2)^k = x^{2k}\). Therefore, to obtain the coefficient of \(x^4\), we need to find the term where \(2k = 4\), which gives \(k = 2\).

3. **Coefficient of \(x^4\)**:
   The term corresponding to \(k = 2\) is:

   \[
   \binom{n+1}{2} 3^{n+1-2} x^{2 \cdot 2} = \binom{n+1}{2} 3^{n-1} x^4
   \]
   
   The coefficient of \(x^4\) is therefore:

   \[
   \binom{n+1}{2} 3^{n-1}
   \]

4. **Setting the Coefficient Equal to 20412**:
   According to the problem, the coefficient of \(x^4\) is given as 20412. Therefore, we have the equation:

   \[
   \binom{n+1}{2} 3^{n-1} = 20412
   \]

   Recall that:

   \[
   \binom{n+1}{2} = \frac{(n+1)n}{2}
   \]

   Substituting this into the equation:

   \[
   \frac{(n+1)n}{2} 3^{n-1} = 20412
   \]

   Multiply both sides by 2:

   \[
   (n+1)n 3^{n-1} = 40824
   \]

5. **Solving for \(n\)**:
   Now we need to solve this equation for \(n\). Testing values of \(n\):

   - For \(n = 12\):
     \[
     (12+1) \cdot 12 \cdot 3^{12-1} = 13 \cdot 12 \cdot 3^{11}
     \]

     Let's compute \(3^{11} = 177147\), so:

     \[
     13 \cdot 12 \cdot 177147 = 27693612
     \]

     This is too large, so we try \(n = 8\).

   - For \(n = 8\):
     \[
     (8+1) \cdot 8 \cdot 3^{8-1} = 9 \cdot 8 \cdot 3^7 = 9 \cdot 8 \cdot 2187 = 157464
     \]

     This is too small, so we'll try adjusting other values systematically,

(3 + x^2)^(n+1) when n is an integer given that the coefficient of x^4 = 20412, find n

Solve for n in the binomial expansion of (3 + x^2)^(n+1) given the coefficient of x^4 is 20412. This problem covers advanced algebraic concepts like binomial expansion and polynomial coefficients, suitable for Grades 11-12 or early college students.

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