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Explanation

Let ` a^(2) (b +c ) , b^(2) (c + a), c^(2) (a + b) ` are in A.P.
`rArr a^(2) b + a^(2) c , b^(2)c + ab^(2) , c^(2) a + bc^(2)` are in A.P.
`rArr a^(2) b + a^(2) c + abc , b^(2) c + ab^(2) + abc, c^(2) a + bc^(2) + abc ` are in A.P. , (adding abc)
` rArr a (ab + ac + bc ) , b( bc + ab + ac) , c (ca + bc + ab) ` are in A.P.
` rArr a, b, c ` are in A.P. which is given to be true , (divide by ` ab + bc + ca)`
Hence ` a^(2) (b + c), b^(2) (c + a) , c^(2) (a + b)` are in A.P.