Question : An example of an equality relation of two expressions in $x$. Which is not an identity, is:
Option 1: $(x+3)^{2}=x^{2}+6x+9$
Option 2: $(x+2y)^{3}=x^{3}+8y^{3}+6xy(x+2y)$
Option 3: $(x+2)^{2}=x^{2}+2x+4$
Option 4: $(x+3)(x–3)=x^{2}–9$
Correct Answer: $(x+2)^{2}=x^{2}+2x+4$
Solution : Given: $(x+3)^{2}=x^{2}+6x+9$ ⇒ $(x^{2}+9+6x)=x^{2}+6x+9$ Similarly, $(x+2y)^{3}=x^{3}+8y^{3}+6xy(x+2y)$ ⇒ $x^{3}+8y^{3}+6xy(x+2y)=x^{3}+8y^{3}+6xy(x+2y)$ Similarly, $(x+3)(x–3)=x^{2}-9$ Similarly, $(x+2)^{2}=x^{2}+2x+4$ ⇒ $(x^{2}+4+4x)=x^{2}+2x+4$ We can see this is not an identity. Hence, the correct answer is '$(x+2)^{2}=x^{2}+2x+4$'.
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