Simplify ( 𝑥 + 1 ) ^2 − ( 𝑥 ^2 + 1 )

The term \((x+1)^{2}\) is expanded using the formula \((a+b)^{2}=a^{2}+2ab+b^{2}\). In this case, \(a=x\) and \(b=1\). Therefore, \((x+1)^{2}=x^{2}+2(x)(1)+1^{2}=x^{2}+2x+1\)The expanded form of \((x+1)^{2}\) is substituted back into the original expression: \(x^{2}+2x+1-(x^{2}+1)\). The negative sign in front of the parenthesis is distributed to each term inside: \(x^{2}+2x+1-x^{2}-1\)Like terms are grouped together and combined: \((x^{2}-x^{2})+2x+(1-1)\). The \(x^{2}\) terms cancel out, and the constant terms cancel out: \(0+2x+0\)The simplified form of the expression \((x+1)^{2}-(x^{2}+1)\) is \(2x\).