\(f(m+n) = (m+n)^2\)
\(f(m-n) = (m-n)^2\)
The above two are special products whose expansions are
\(f(m+n) = (m+n)^2 = m^2 + n^2 + 2mn
\)
\(f(m-n) = (m-n)^2 = m^2 + n^2 - 2mn\)
\(f(m+n) + f(m-n) = (m+n)^2 + (m-n)^2 = m^2 + n^2 + 2mn + m^2 + n^2 - 2mn\)
\(f(m+n) + f(m-n) = 2m^2 + 2n^2\)
Hence answer is
Choice C.
Note: The three special products which you should understand and memorize are:\((a+b)^2 = a^2 + b^2 + 2ab\)
\((a-b)^2 = a^2 + b^2 - 2ab\)
\((a+b)(a-b) = a^2 - b^2\)
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