If ` ( x - a^(2))/( b + c) +( x - b^(2))/( c + a) + (x - c^(2))/( a + b) = 4 (a + b+ c)` the x is equal to

Options

A. A)`(a + b + c)^(2)`
B. B)`a^(2) + b^(2) + c^(2)`
C. C)`ab + bc + ca`
D. D)`a^(2) + b^(2) + c^(2) - ab - bc - ca `

Answer

To solve the equation

\[ \frac{x - a^2}{b + c} + \frac{x - b^2}{c + a} + \frac{x - c^2}{a + b} = 4(a + b + c), \]

we can follow these steps:

### Step 1: Substitute values for \( a, b, c \)

Let's assume \( a = b = c = k \). This simplifies our calculations and helps us understand the structure of the equation.

### Step 2: Substitute into the equation

Substituting \( a, b, c \) with \( k \):

\[ \frac{x - k^2}{k + k} + \frac{x - k^2}{k + k} + \frac{x - k^2}{k + k} = 4(3k). \]

This simplifies to:

\[ \frac{x - k^2}{2k} + \frac{x - k^2}{2k} + \frac{x - k^2}{2k} = 12k. \]

### Step 3: Combine the fractions

Since we have three identical fractions, we can combine them:

\[ 3 \cdot \frac{x - k^2}{2k} = 12k. \]

### Step 4: Simplify the equation

This simplifies to:

\[ \frac{3(x - k^2)}{2k} = 12k. \]

### Step 5: Multiply both sides by \( 2k \)

To eliminate the fraction, multiply both sides by \( 2k \):

\[ 3(x - k^2) = 24k^2. \]

### Step 6: Distribute and isolate \( x \)

Distributing the 3 gives:

\[ 3x - 3k^2 = 24k^2. \]

Now, isolate \( x \):

\[ 3x = 24k^2 + 3k^2. \]

This simplifies to:

\[ 3x = 27k^2. \]

### Step 7: Solve for \( x \)

Finally, divide both sides by 3:

\[ x = 9k^2. \]

### Step 8: Express in terms of \( a, b, c \)

Since \( k = \frac{a + b + c}{3} \), we can express \( x \) in terms of \( a, b, c \):

\[ x = 9\left(\frac{a + b + c}{3}\right)^2 = \frac{(a + b + c)^2}{3}. \]

### Conclusion

Thus, the value of \( x \) is

\[ x = 9k^2 \text{ or } x = (a + b + c)^2. \]