If a=2012, `b=-1005, c=-1007`, then the value of `a^(4)/(b+c) +b^(4)/(c+a) + c^(4)/(a+b) + 3abc` is:

Answer

To solve the problem, we need to evaluate the expression:

\[ \frac{a^4}{b+c} + \frac{b^4}{c+a} + \frac{c^4}{a+b} + 3abc \]

Given values are:
- \( a = 2012 \)
- \( b = -1005 \)
- \( c = -1007 \)

### Step 1: Calculate \( b+c \), \( c+a \), and \( a+b \)

First, we calculate \( b+c \), \( c+a \), and \( a+b \):

\[ b+c = -1005 + (-1007) = -2012 \]
\[ c+a = -1007 + 2012 = 1005 \]
\[ a+b = 2012 - 1005 = 1007 \]

### Step 2: Substitute values into the expression

Now, substitute these values into the expression:

\[ \frac{a^4}{b+c} + \frac{b^4}{c+a} + \frac{c^4}{a+b} + 3abc = \frac{2012^4}{-2012} + \frac{(-1005)^4}{1005} + \frac{(-1007)^4}{1007} + 3(2012)(-1005)(-1007) \]

### Step 3: Simplify each term

1. **First term:**
\[ \frac{2012^4}{-2012} = -2012^3 \]

2. **Second term:**
\[ \frac{(-1005)^4}{1005} = 1005^3 \]

3. **Third term:**
\[ \frac{(-1007)^4}{1007} = 1007^3 \]

4. **Fourth term:**
\[ 3(2012)(-1005)(-1007) = 3 \cdot 2012 \cdot 1005 \cdot 1007 \]

### Step 4: Combine the terms

Now, we combine the terms:

\[ -2012^3 + 1005^3 + 1007^3 + 3 \cdot 2012 \cdot 1005 \cdot 1007 \]

### Step 5: Use the identity for cubes

We can use the identity for the sum of cubes:

\[ x^3 + y^3 + z^3 - 3xyz = (x+y+z)(x^2 + y^2 + z^2 - xy - xz - yz) \]

Let \( x = 1005 \), \( y = 1007 \), and \( z = -2012 \).

Calculating \( x+y+z \):

\[ 1005 + 1007 - 2012 = 0 \]

Since \( x + y + z = 0 \), we have:

\[ x^3 + y^3 + z^3 - 3xyz = 0 \]

Thus,

\[ x^3 + y^3 + z^3 = 3xyz \]

### Step 6: Calculate \( 3xyz \)

Now we calculate \( 3xyz \):

\[ 3xyz = 3 \cdot 1005 \cdot 1007 \cdot (-2012) \]

### Final Result

Thus, the entire expression evaluates to:

\[ 0 \]