Answer:[tex]=\frac{a+5-4\sqrt{a+1}}{a-3}[/tex]Step-by-step explanation:Given:[tex]\frac{\sqrt{a+1}-2}{\sqrt{a+1}+2}[/tex]Rationalise the denominator. Solution:Simplify the expression.[tex]=\frac{\sqrt{a+1} -2}{\sqrt{a+1} +2 }[/tex]Multiply numerator by both denominator and numerator.[tex]=\frac{\sqrt{a+1} -2}{\sqrt{a+1}+2}\times \frac{\sqrt{a+1}-2}{\sqrt{a+1}-2}[/tex][tex]=\frac{(\sqrt{a+1}-2)(\sqrt{a+1}-2)}{(\sqrt{a+1}+2)(\sqrt{a+1}-2)}[/tex]Assume [tex]\sqrt{a+1} =a\ and\ 2=b[/tex]Applying formula [tex](a-b)(a+b)=a^{2} -b^{2}[/tex], so we get[tex]=\frac{(\sqrt{a+1} -2)^{2}}{(\sqrt{a+1})^{2} -2^{2})}[/tex][tex]=\frac{(\sqrt{a+1})^{2} +2^{2}-2(\sqrt{a+1})(2)}{(a+1)-4}[/tex][tex]=\frac{(a+1) +4-4\sqrt{a+1}}{a+1-4}[/tex][tex]=\frac{a+5-4\sqrt{a+1}}{a-3}[/tex]Therefore, the simplification of the expression is given below.[tex]=\frac{a+5-4\sqrt{a+1}}{a-3}[/tex]