Sometimes by adding or subtracting a term we may get a perfect cube or a perfect square, so in the present case we can take
x^3 + 3x^2 + 3x - 7
= x^3 + 3x^2 + 3x + 1 - 1 - 7 ,by adding and subtracting 1 ,
so that the value of the expression does not change
= x^3 + 3x^2 + 3x + 1 - 8
= (x^3 + 3x^2 + 3x +1) - 8, by associativity
= (x+1)^3 - 2^3, since (A+B)^3 = A^3 + 3A^2B + 3AB^2 + B^3
= (x + 1 - 2) [ (x+1)^2 + (x+1)2 + 2^2 ], since A^3 - B^3 = (A-B) (A^2 + AB + B^2)
= (x - 1) ( x^2 + 2x + 1 + 2x +2 +4) , using (A + B) ^2 = A^2 + 2AB + B^2
= (x - 1) (x^2 + 4x + 7) Ans., adding like terms
x^3 + 3x^2 + 3x - 7
= x^3 + 3x^2 + 3x + 1 - 1 - 7 ,by adding and subtracting 1 ,
so that the value of the expression does not change
= x^3 + 3x^2 + 3x + 1 - 8
= (x^3 + 3x^2 + 3x +1) - 8, by associativity
= (x+1)^3 - 2^3, since (A+B)^3 = A^3 + 3A^2B + 3AB^2 + B^3
= (x + 1 - 2) [ (x+1)^2 + (x+1)2 + 2^2 ], since A^3 - B^3 = (A-B) (A^2 + AB + B^2)
= (x - 1) ( x^2 + 2x + 1 + 2x +2 +4) , using (A + B) ^2 = A^2 + 2AB + B^2
= (x - 1) (x^2 + 4x + 7) Ans., adding like terms
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