
Steps  Solution  
1.  Assume an expression of the form (ax+b)^{n}, considering the power of x, that comes nearly as the expression.  In this case, let us assume (x+2)^{2}  
2.  Compare them, to check the deficient terms.  In this case, we get: (x+2)^{2} = x^{2} + 4x + 4 And, the expression to solve: x^{2} + 3x + 2 Clearly, the deficient terms are: x + 2 

3.  Add the deficient terms to both side of the equation, to complete the assumed expression.  In this case, adding (x + 2): (x^{2} + 3x + 2) + (x + 2) = 0 + (x + 2) or, x^{2} + 4x + 4 = (x + 2) or, (x + 2)^{2} = (x + 2) 

4.  Replace common terms with another variable, and solve.  In this case, let y = (x + 2). Then we get: y^{2} = y This solves, y = 0 and y = 1 

5.  Use solution to solve original variable.  In this case, y = (x + 2). y = 0, gives: x + 2 = 0; or, x = 2 And, y = 1, gives: x + 2 = 1; or, x = 1 Thus, we get: x = 2, x = 1, which is the answer! 
Let us assume (x+2)^{3}: (x + 2)^{3} = x^{3} + 6x^{2} + 12x + 8 Clearly, the deficient terms are: (x + 2) Adding to both sides: (x^{3} + 6x^{2} + 11x + 6) + (x + 2) = 0 + (x + 2) or, (x + 2)^{3} = (x + 2) Let, y = (x + 2). Then, y^{3} = y solves for 0, 1 and 1 x + 2 = 0 gives: x = 2 Also, x + 2 = 1 gives: x = 1 And, x + 2 = 1 gives: x = 3 Thus, we get: x = 2, x = 1, x = 3 
Let us assume (x+3)^{3}: (x + 3)^{3} = x^{3} + 9x^{2} + 27x + 27 Clearly, the deficient terms are: (x^{2} + 10x + 17) Adding to both sides: (x^{3} + 8x^{2} + 17x + 10) + (x^{2} + 10x + 17) = 0 + (x^{2} + 10x + 17) or, (x + 3)^{3} = (x^{2} + 10x + 17) Let, y = (x + 3). Then, y^{3} = y^{2} + 4y – 4 This solves for 1, 2 and 2 x + 3 = 1 gives: x = 2 Also, x + 3 = 2 gives: x = 1 And, x + 3 = 2 gives: x = 5 Thus, we get: x = 2, x = 1, x = 5 
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