Find the values of the following determinants
`|{:(12,-10,5),(3,2,-1),(-4,0,3):}|`

Answer

To find the value of the determinant

\[ D = \begin{vmatrix} 12 & -10 & 5 \\ 3 & 2 & -1 \\ -4 & 0 & 3 \end{vmatrix} \]

we will use the method of cofactor expansion along the first row.

### Step 1: Write the determinant using cofactor expansion

Using the first row, we have:

\[ D = 12 \cdot \begin{vmatrix} 2 & -1 \\ 0 & 3 \end{vmatrix} - (-10) \cdot \begin{vmatrix} 3 & -1 \\ -4 & 3 \end{vmatrix} + 5 \cdot \begin{vmatrix} 3 & 2 \\ -4 & 0 \end{vmatrix} \]

### Step 2: Calculate the 2x2 determinants

1. For the first determinant:

\[ \begin{vmatrix} 2 & -1 \\ 0 & 3 \end{vmatrix} = (2 \cdot 3) - (0 \cdot -1) = 6 \]

2. For the second determinant:

\[ \begin{vmatrix} 3 & -1 \\ -4 & 3 \end{vmatrix} = (3 \cdot 3) - (-1 \cdot -4) = 9 - 4 = 5 \]

3. For the third determinant:

\[ \begin{vmatrix} 3 & 2 \\ -4 & 0 \end{vmatrix} = (3 \cdot 0) - (2 \cdot -4) = 0 + 8 = 8 \]

### Step 3: Substitute back into the determinant formula

Now substituting back into the expression for \(D\):

\[ D = 12 \cdot 6 + 10 \cdot 5 + 5 \cdot 8 \]

### Step 4: Calculate the final value

Calculating each term:

1. \(12 \cdot 6 = 72\)
2. \(10 \cdot 5 = 50\)
3. \(5 \cdot 8 = 40\)

Now, summing these values:

\[ D = 72 + 50 + 40 = 162 \]

### Final Answer

Thus, the value of the determinant is

\[ \boxed{162} \]