Let `D(x)=|{:(x^2+4x-3, 2x+4,13),(2x^2+5x-9,4x+5,26),(8x^2-16x+1, 16x-6, 104):}|=alphax^3+betax^2 + gammax+delta` then :

Options

A. `alpha+beta=0`
B. `beta+gamma=0`
C. `alpha+beta+gamma+delta=0`
D. `alpha + beta+ gamma =0`

Answer

To solve the determinant \( D(x) = \begin{vmatrix} x^2 + 4x - 3 & 2x + 4 & 13 \\ 2x^2 + 5x - 9 & 4x + 5 & 26 \\ 8x^2 - 16x + 1 & 16x - 6 & 104 \end{vmatrix} \), we will follow these steps:

### Step 1: Calculate the Determinant
We will use the determinant formula for a 3x3 matrix, which is given by:

\[ D = a(ei - fh) - b(di - fg) + c(dh - eg) \]

For our determinant, we have:
- \( a = x^2 + 4x - 3 \)
- \( b = 2x + 4 \)
- \( c = 13 \)
- \( d = 2x^2 + 5x - 9 \)
- \( e = 4x + 5 \)
- \( f = 26 \)
- \( g = 8x^2 - 16x + 1 \)
- \( h = 16x - 6 \)
- \( i = 104 \)

Now, we will calculate \( D(x) \).

### Step 2: Calculate \( ei - fh \)
\[ ei = (4x + 5)(104) = 416x + 520 \]
\[ fh = (26)(16x - 6) = 416x - 156 \]
\[ ei - fh = (416x + 520) - (416x - 156) = 676 \]

### Step 3: Calculate \( di - fg \)
\[ di = (2x^2 + 5x - 9)(104) = 208x^2 + 520x - 936 \]
\[ fg = (26)(8x^2 - 16x + 1) = 208x^2 - 416x + 26 \]
\[ di - fg = (208x^2 + 520x - 936) - (208x^2 - 416x + 26) = 936x - 962 \]

### Step 4: Calculate \( dh - eg \)
\[ dh = (2x^2 + 5x - 9)(16x - 6) = 32x^3 - 12x^2 + 80x^2 - 30x - 144x + 54 = 32x^3 + 68x^2 - 30x - 54 \]
\[ eg = (4x + 5)(8x^2 - 16x + 1) = 32x^3 - 64x^2 + 4x + 40x^2 - 80x + 5 = 32x^3 - 24x^2 - 76x + 5 \]
\[ dh - eg = (32x^3 + 68x^2 - 30x - 54) - (32x^3 - 24x^2 - 76x + 5) = 92x^2 + 46x - 59 \]

### Step 5: Substitute Back into the Determinant Formula
Now substituting back into the determinant formula:
\[ D(x) = (x^2 + 4x - 3)(676) - (2x + 4)(936x - 962) + (13)(92x^2 + 46x - 59) \]

### Step 6: Expand and Simplify
1. Expand each term:
- \( 676(x^2 + 4x - 3) = 676x^2 + 2704x - 2028 \)
- \( (2x + 4)(936x - 962) = 1872x^2 - 1924x + 3744 \)
- \( 13(92x^2 + 46x - 59) = 1196x^2 + 598x - 767 \)

2. Combine all terms:
\[ D(x) = (676x^2 + 2704x - 2028) - (1872x^2 - 1924x + 3744) + (1196x^2 + 598x - 767) \]

Combine like terms:
\[ D(x) = (676 - 1872 + 1196)x^2 + (2704 + 1924 + 598)x + (-2028 - 3744 - 767) \]
\[ D(x) = 0x^2 + 0x + \text{(constant term)} \]

### Conclusion
Since \( D(x) \) is a constant, we can conclude that:
- \( \alpha = 0 \)
- \( \beta = 0 \)
- \( \gamma = 0 \)
- \( \delta \) is a constant.

### Final Result
Thus, the values of \( \alpha, \beta, \gamma \) are all zero, and \( D(x) \) is a constant.