Study the given statements.
Statement I: `(a)/(x + a) + (b)/( x -a) - (c)/( x ^(2) - a ^(2))= (a (x-a) + b (x +a) - c)/(x ^(2) - a ^(2))`
Statement II: `3 abc - a ^(3) - b ^(3) - c ^(3) = (1)/(2) (a + b + c) [(a - b) ^(2) + (b -c) ^(2) + (c -a) ^(2) ]`
Which of the following options holds ?

Options

A. Both Statement I and Statement II are true.
B. Statement I is true and Statement II is false.
C. Statement I is false false and Statement II is true.
D. Both Statement I and Statement II are false.

Answer

To solve the given problem, we will analyze both statements step by step.

### Statement I:
We need to verify if:
\[ \frac{a}{x + a} + \frac{b}{x - a} - \frac{c}{x^2 - a^2} = \frac{a(x - a) + b(x + a) - c}{x^2 - a^2} \]

**Step 1: Find a common denominator for the left-hand side (LHS).**
The common denominator for the fractions is \( (x + a)(x - a) = x^2 - a^2 \).

**Step 2: Rewrite each term with the common denominator.**
\[ \frac{a}{x + a} = \frac{a(x - a)}{(x + a)(x - a)} = \frac{a(x - a)}{x^2 - a^2} \]
\[ \frac{b}{x - a} = \frac{b(x + a)}{(x - a)(x + a)} = \frac{b(x + a)}{x^2 - a^2} \]
\[ -\frac{c}{x^2 - a^2} = -\frac{c}{x^2 - a^2} \]

**Step 3: Combine the fractions.**
\[ \frac{a(x - a) + b(x + a) - c}{x^2 - a^2} \]

**Step 4: Compare LHS and RHS.**
Both sides simplify to:
\[ \frac{a(x - a) + b(x + a) - c}{x^2 - a^2} \]
Thus, Statement I is **True**.

### Statement II:
We need to verify if:
\[ 3abc - a^3 - b^3 - c^3 = \frac{1}{2}(a + b + c)\left[(a - b)^2 + (b - c)^2 + (c - a)^2\right] \]

**Step 1: Expand the right-hand side (RHS).**
Using the identity \( (a - b)^2 = a^2 - 2ab + b^2 \):
\[ (a - b)^2 + (b - c)^2 + (c - a)^2 = (a^2 - 2ab + b^2) + (b^2 - 2bc + c^2) + (c^2 - 2ca + a^2) \]
Combining like terms:
\[ = 2a^2 + 2b^2 + 2c^2 - 2(ab + bc + ca) \]

**Step 2: Substitute back into RHS.**
\[ RHS = \frac{1}{2}(a + b + c)(2a^2 + 2b^2 + 2c^2 - 2(ab + bc + ca)) \]
\[ = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca) \]

**Step 3: Expand the expression.**
This will yield:
\[ = a^3 + b^3 + c^3 - 3abc \]

**Step 4: Compare LHS and RHS.**
LHS is \( 3abc - a^3 - b^3 - c^3 \) and RHS simplifies to \( a^3 + b^3 + c^3 - 3abc \), which means:
\[ 3abc - a^3 - b^3 - c^3 \neq a^3 + b^3 + c^3 - 3abc \]
Thus, Statement II is **False**.

### Conclusion:
- Statement I is **True**.
- Statement II is **False**.

### Final Answer:
The correct option is that Statement I is true and Statement II is false.