Để pt có 2 nghiệm \(\Leftrightarrow\left\{{}\begin{matrix}m+1\ne0\\\Delta'=\left(m+2\right)^2-\left(m-3\right)\left(m+1\right)\ge0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}m\ne-1\\6m+7\ge0\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}m\ne-1\\m\ge-\frac{7}{6}\end{matrix}\right.\)
Theo Viet: \(\left\{{}\begin{matrix}x_1+x_2=\frac{2m+4}{m+1}\\x_1x_2=\frac{m-3}{m+1}\end{matrix}\right.\)
\(\left(4x_1+1\right)\left(4x_2+1\right)=18\)
\(\Leftrightarrow16x_1x_2+4\left(x_1+x_2\right)-17=0\)
\(\Leftrightarrow\frac{16\left(m-3\right)}{m+1}+\frac{8\left(m+2\right)}{m+1}-17=0\)
\(\Leftrightarrow16\left(m-3\right)+8\left(m+2\right)-17\left(m+1\right)=0\)
\(\Rightarrow7m-49=0\Rightarrow m=7\)
b/ \(\left\{{}\begin{matrix}2x_1+2x_2=\frac{4m+8}{m+1}\\x_1x_2=\frac{m-3}{m+1}\end{matrix}\right.\)
\(\Leftrightarrow2x_1+2x_2+x_1x_2=\frac{5m+5}{m+1}\)
\(\Leftrightarrow2x_1+2x_2+x_1x_2=5\)
Đây là biểu thức liên hệ 2 nghiệm ko phụ thuộc m