[tex]\displaystyle\bf\\\frac{\sqrt{a-4\sqrt{a-1} +3} }{\sqrt{a-1} -2} +1=\frac{\sqrt{(a-1)-4\sqrt{a-1} +4} }{\sqrt{a-1} -2}+1 =\\\\\\=\frac{\sqrt{(\sqrt{a-1} )^{2} -2\cdot\sqrt{a-1}\cdot 2 +2^{2} } }{\sqrt{a-1} -2} +1=\frac{\sqrt{(\sqrt{a-1} -2)^{2} } }{\sqrt{a-1} -2}+1 =\\\\\\=\frac{\sqrt{a-1} -2}{\sqrt{a-1} -2} +1=1+1=2\\\\\\Otvet \ : \ 2[/tex]