Lời giải:
a)
\(x^4+2x^3+5x^2+4x-12=0\)
\(\Leftrightarrow x^3(x-1)+3x^2(x-1)+8x(x-1)+12(x-1)=0\)
\(\Leftrightarrow (x-1)(x^3+3x^2+8x+12)=0\)
\(\Leftrightarrow (x-1)[x^2(x+2)+x(x+2)+6(x+2)]=0\)
\(\Leftrightarrow (x-1)(x+2)(x^2+x+6)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x=1\\x=-2\\x^2+x+6=0\left(1\right)\end{matrix}\right.\)
Đối với (1): \(\Leftrightarrow (x+\frac{1}{2})^2+\frac{23}{4}=0\)
(vô lý vì \((x+\frac{1}{2})^2+\frac{23}{4}\geq \frac{23}{4}>0\) )
Do đó \(x\in\left\{-2;1\right\}\)
b) ĐKXĐ: ......
\(\frac{1}{x^2+4x+3}+\frac{1}{x^2+8x+15}=\frac{1}{6}\)
\(\Leftrightarrow \frac{1}{(x+1)(x+3)}+\frac{1}{(x+3)(x+5)}=\frac{1}{6}\)
\(\Leftrightarrow \frac{(x+5)+(x+1)}{(x+1)(x+3)(x+5)}=\frac{1}{6}\)
\(\Leftrightarrow \frac{2(x+3)}{(x+1)(x+3)(x+5)}=\frac{1}{6}\Leftrightarrow \frac{2}{(x+1)(x+5)}=\frac{1}{6}\)
\(\Leftrightarrow (x+1)(x+5)=12\)
\(\Leftrightarrow x^2+6x-7=0\)
\(\Leftrightarrow (x-1)(x+7)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x=1\\x=-7\end{matrix}\right.\) (thỏa mãn đkxđ)
Vậy \(x\in\left\{-7;1\right\}\)