If the expression `|{:(x^2+x+3,1,4),(2x^4+x^3+2x+1,2,3),(x^2+x,1,1):}|` is equal to `ax^4+bx^3+cx^2+dx+e`, then the value of e is equal to

Options

A. zero
B. 1
C. 2
D. none of these

Answer

To solve the determinant given in the expression \( |{:(x^2+x+3,1,4),(2x^4+x^3+2x+1,2,3),(x^2+x,1,1):}| \) and find the value of \( e \) in the polynomial \( ax^4 + bx^3 + cx^2 + dx + e \), we can follow these steps:

### Step 1: Write the Determinant
We need to evaluate the determinant:
\[ D = \begin{vmatrix} x^2 + x + 3 & 1 & 4 \\ 2x^4 + x^3 + 2x + 1 & 2 & 3 \\ x^2 + x & 1 & 1 \end{vmatrix} \]

### Step 2: Expand the Determinant
We can expand the determinant using the first row:
\[ D = (x^2 + x + 3) \begin{vmatrix} 2 & 3 \\ 1 & 1 \end{vmatrix} - 1 \begin{vmatrix} 2x^4 + x^3 + 2x + 1 & 3 \\ x^2 + x & 1 \end{vmatrix} + 4 \begin{vmatrix} 2x^4 + x^3 + 2x + 1 & 2 \\ x^2 + x & 1 \end{vmatrix} \]

### Step 3: Calculate the 2x2 Determinants
1. For the first determinant:
\[ \begin{vmatrix} 2 & 3 \\ 1 & 1 \end{vmatrix} = (2 \cdot 1) - (3 \cdot 1) = 2 - 3 = -1 \]

2. For the second determinant:
\[ \begin{vmatrix} 2x^4 + x^3 + 2x + 1 & 3 \\ x^2 + x & 1 \end{vmatrix} = (2x^4 + x^3 + 2x + 1) \cdot 1 - (3 \cdot (x^2 + x)) = 2x^4 + x^3 + 2x + 1 - 3x^2 - 3x = 2x^4 + x^3 - 3x^2 - x + 1 \]

3. For the third determinant:
\[ \begin{vmatrix} 2x^4 + x^3 + 2x + 1 & 2 \\ x^2 + x & 1 \end{vmatrix} = (2x^4 + x^3 + 2x + 1) \cdot 1 - (2 \cdot (x^2 + x)) = 2x^4 + x^3 + 2x + 1 - 2x^2 - 2x = 2x^4 + x^3 - 2x^2 + 1 \]

### Step 4: Substitute Back into the Determinant
Now substitute these back into the determinant:
\[ D = (x^2 + x + 3)(-1) - 1(2x^4 + x^3 - 3x^2 - x + 1) + 4(2x^4 + x^3 - 2x^2 + 1) \]
\[ D = -(x^2 + x + 3) - (2x^4 + x^3 - 3x^2 - x + 1) + 4(2x^4 + x^3 - 2x^2 + 1) \]

### Step 5: Simplify the Expression
Now, simplify the expression:
\[ D = -x^2 - x - 3 - 2x^4 - x^3 + 3x^2 + x - 1 + 8x^4 + 4x^3 - 8x^2 + 4 \]
Combine like terms:
\[ D = (8x^4 - 2x^4) + (-x^3 + 4x^3) + (-x^2 + 3x^2 - 8x^2) + (-3 - 1 + 4) \]
\[ D = 6x^4 + 3x^3 - 6x^2 + 0 \]

### Step 6: Identify Coefficients
From the expression \( D = 6x^4 + 3x^3 - 6x^2 + 0 \), we can see that:
- \( a = 6 \)
- \( b = 3 \)
- \( c = -6 \)
- \( d = 0 \)
- \( e = 0 \)

### Conclusion
Thus, the value of \( e \) is:
\[ \boxed{0} \]