ĐKXĐ: \(x\ge2\)

Khi đó ta có \(x^2-x+1\ge3\Rightarrow1-2\sqrt{x^2-x+1}< 0\)

Do đó BPT tương đương:

\(\sqrt{2\left(x^2+7x+3\right)}-\sqrt{x^2+x-6}-3\sqrt{x+1}\le0\)

\(\Leftrightarrow\sqrt{2x^2+14x+6}\le\sqrt{x^2+x-6}+3\sqrt{x+1}\)

\(\Leftrightarrow2x^2+14x+6\le x^2+10x+3+6\sqrt{\left(x+1\right)\left(x^2+x-6\right)}\)

\(\Leftrightarrow x^2+4x+3\le6\sqrt{\left(x+1\right)\left(x+3\right)\left(x-2\right)}\)

\(\Leftrightarrow\left(x+1\right)\left(x+3\right)\le6\sqrt{\left(x+1\right)\left(x+3\right)\left(x-2\right)}\)

\(\Leftrightarrow\sqrt{\left(x+1\right)\left(x+3\right)}\le6\sqrt{x-2}\)

\(\Leftrightarrow\left(x+1\right)\left(x+3\right)\le36\left(x-2\right)\)

\(\Leftrightarrow x^2-32x+75\le0\)

\(\Rightarrow16-\sqrt{181}\le x\le16+\sqrt{181}\)

a: =>\(\dfrac{x^2+2x-13-x+1}{x-1}< 0\)

=>\(\dfrac{x^2+x-12}{x-1}< 0\)

=>\(\dfrac{\left(x+4\right)\left(x-3\right)}{x-1}< 0\)

=>1<x<3 hoặc x<-4

b: =>\(\dfrac{3x^2+4x-3x-4}{x-1}< 3\)

=>3x+4<3

=>3x<-1

=>x<-1/3

c: TH1: 2x^2-3x+1>0 và x+2>0

=>(2x-1)(x-1)>0 và x+2>0

=>x>1

TH2: (2x-1)(x-1)<0 và x+2<0

=>x<-2 và 1/2<x<1

=>Loại