ĐK: \(x\ne1\)
\(M=\frac{4\left(x^2+x+1\right)}{4\left(x^2+2x+1\right)}=\frac{3x^2+6x+3+x^2-2x+1}{4\left(x+1\right)^2}=\frac{3\left(x+1\right)^2+\left(x-1\right)^2}{4\left(x+1\right)^2}\)
\(M=\frac{3}{4}+\frac{\left(x-1\right)^2}{4\left(x+1\right)^2}\ge\frac{3}{4}\)
\(\Rightarrow M_{min}=\frac{3}{4}\) khi \(\frac{\left(x-1\right)^2}{4\left(x+1\right)^2}=0\Rightarrow x=1\)
ĐK: \(x\ne-1\)
\(\Rightarrow Mx^2+2Mx+M-x^2-x-1=0\)
\(\Leftrightarrow\left(M-1\right)x^2+\left(2M-1\right)x+M-1=0\)
Để pt có nghiệm thì
\(\Delta\ge0\)
\(\Rightarrow\left(2M-1\right)^2-4\left(M-1\right)^2\ge0\)
\(\Leftrightarrow-4M+1-4\left(-2M+1\right)\ge0\)
\(\Leftrightarrow2M-3\ge0\)
\(\Leftrightarrow M\ge\frac{3}{2}\)
Mmin\(=\frac{3}{2}\Leftrightarrow\frac{x^2+x+1}{\left(x+1\right)^2}=\frac{3}{2}\)
\(\Rightarrow2x^2+2x+2-3x^2-6x-3=0\)
\(\Leftrightarrow-x^2-4x-1=0\)
\(\Leftrightarrow x=-2\pm\sqrt{3}\left(TM\right)\)