<s>[INST] Proving an algebraic identity Prove:
$$(a + b + c)(ab + bc + ca) - abc = (a + b)(b + c)(c + a)$$
Problem:
I am not sure how to proceed after expanding the brackets on the RHS. I am not sure if I also expanded correctly. My solution is:
 [/INST] $$(a+b)(b+c)(c+a)$$
$$=(ab+bc+ca+b^2)(c+a)$$
$$=(ab+bc+ca)(a+c)+ab^2+b^2c$$
Adding $abc$ gives:
$$(ab+bc+ca)(a+c)+ab^2+b^2c+abc$$
$$=(ab+bc+ca)(a+c)+b(ab+bc+ac)$$
$$=(ab+bc+ca)(a+b+c)$$
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