\(a.\sqrt{x+3}+\sqrt{3x+1}=2\sqrt{x}+\sqrt{2x+2}\)
\(\Leftrightarrow\left(\sqrt{3x+1}-\sqrt{4x}\right)+\left(\sqrt{x+3}-\sqrt{2x+2}\right)=0\)
\(\Leftrightarrow\frac{3x+1-4x}{\sqrt{3x+1}+\sqrt{4x}}+\frac{x+3-2x-2}{\sqrt{x+3}+\sqrt{2x+2}}=0\)
\(\Leftrightarrow\frac{-x+1}{\sqrt{3x+1}+\sqrt{4x}}+\frac{-x+1}{\sqrt{x+3}+\sqrt{2x+2}}=0\)
\(\left(1-x\right)\left(\frac{1}{\sqrt{x+3}+\sqrt{2x+2}}+\frac{1}{\sqrt{3x+1}+\sqrt{4x}}\right)=0\)
\(\Rightarrow x=1\)
\(\sqrt{\frac{x^3+1}{x+1}}+\sqrt{x+1}=\sqrt{x^2-x+1}+\sqrt{x+3}\left(x>-1\right)\)
\(\Rightarrow\sqrt{x^3+1}+\sqrt{\left(x+1\right)^2}=\sqrt{x^3+1}+\sqrt{\left(x+3\right)\left(x+1\right)}\)
\(\Leftrightarrow\left(x+1\right)^2=\left(x+3\right)\left(x+1\right)\)
\(\Leftrightarrow x^2+2x+1=x^2+4x+3\)
\(\Leftrightarrow x=-1\)(vô lý)
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