To solve the problem, we need to simplify the given expression step by step:

Step 1: Expand the numerator

The numerator is ((a + b + c)^2 - (a - b - c)^2). Using the difference of squares formula (x^2 - y^2 = (x - y)(x + y)):
Let (x = a + b + c) and (y = a - b - c):
[ x - y = (a + b + c) - (a - b - c) = 2b + 2c = 2(b + c) \
x + y = (a + b + c) + (a - b - c) = 2a \
\text{Numerator} = (2(b + c))(2a) = 4a(b + c) ]

Step 2: Expand the denominator

The denominator is ((b + c)^2 - (b - c)^2). Again using the difference of squares formula:
Let (x = b + c) and (y = b - c):
[ x - y = (b + c) - (b - c) = 2c \
x + y = (b + c) + (b - c) = 2b \
\text{Denominator} = (2c)(2b) = 4bc ]

Step 3: Simplify the fraction

[ \frac{\text{Numerator}}{\text{Denominator}} = \frac{4a(b + c)}{4bc} = \frac{a(b + c)}{bc} = \frac{a}{b} + \frac{a}{c} ]

Answer: (\boxed{\dfrac{a}{b} + \dfrac{a}{c}}) (or equivalently (\boxed{\dfrac{a(b + c)}{bc}}))

Since the problem asks for a simplified form, the expanded version (\frac{a}{b} + \frac{a}{c}) is likely the intended answer.

(\boxed{\dfrac{a}{b} + \dfrac{a}{c}})

作者声明:本文包含人工智能生成内容。