((𝒂+𝒃+𝒄)^𝟐−(𝒂+𝒃−𝒄)^𝟐 )/𝒃𝒄+𝒄𝒂

1. Expand both squares in the numerator:

(a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca

(a+b-c)^2 = a^2 + b^2 + c^2 + 2ab - 2bc - 2ca

2. Subtract the two expanded forms:

(a+b+c)^2 - (a+b-c)^2 = (a^2 + b^2 + c^2 + 2ab + 2bc + 2ca) - (a^2 + b^2 + c^2 + 2ab - 2bc - 2ca)

= (a^2 + b^2 + c^2 + 2ab + 2bc + 2ca) - (a^2 + b^2 + c^2 + 2ab - 2bc - 2ca)

= 4bc + 4ca

3. Therefore, the numerator simplifies to:

4bc + 4ca

4. The whole expression becomes:

\frac{4bc + 4ca}{bc + ca}

5. This simplifies to:

\frac{4(bc + ca)}{bc + ca} = 4