Let `a,b,c , in R` not all are equal and `Delta_(1)= |{:(a,,b,,c),(b,,c,,a),(c,,a,,b):}|`
`Delta_(2)= |{:(a+2b,,b+3c,,c+4a),(b+2c,,c+3a,,a+4b),(c+2a,,a+3b,,b+4c):}|" then " (Delta_(2))/(Delta_(1)) = "____"`

Explanation

To solve the problem, we need to compute the ratio \(\frac{\Delta_2}{\Delta_1}\) where:

\[ \Delta_1 = \begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix} \]

and

\[ \Delta_2 = \begin{vmatrix} a + 2b & b + 3c & c + 4a \\ b + 2c & c + 3a & a + 4b \\ c + 2a & a + 3b & b + 4c \end{vmatrix} \]

### Step 1: Express \(\Delta_2\) in terms of \(\Delta_1\)

We can split \(\Delta_2\) into two parts:

\[ \Delta_2 = \begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix} + \begin{vmatrix} 2b & 3c & 4a \\ 2c & 3a & 4b \\ 2a & 3b & 4c \end{vmatrix} \]

The first determinant is \(\Delta_1\).

### Step 2: Factor out constants from the second determinant

In the second determinant, we can factor out constants from each column:

\[ \Delta_2 = \Delta_1 + 2 \begin{vmatrix} b & c & a \\ c & a & b \\ a & b & c \end{vmatrix} + 3 \begin{vmatrix} b & c & a \\ c & a & b \\ a & b & c \end{vmatrix} + 4 \begin{vmatrix} b & c & a \\ c & a & b \\ a & b & c \end{vmatrix} \]

### Step 3: Combine the determinants

Now, we can combine these determinants:

\[ \Delta_2 = \Delta_1 + 2 \cdot 1 + 3 \cdot 1 + 4 \cdot 1 = \Delta_1 + (2 + 3 + 4) \Delta_1 = \Delta_1 + 9 \Delta_1 = 10 \Delta_1 \]

### Step 4: Calculate the ratio

Now we can find the ratio:

\[ \frac{\Delta_2}{\Delta_1} = \frac{10 \Delta_1}{\Delta_1} = 10 \]

### Final Answer

Thus, the value of \(\frac{\Delta_2}{\Delta_1}\) is:

\[ \boxed{10} \]