a)\(\left( {2x + 1} \right)\left( {3x - 2} \right) = \left( {5x - 8} \right)\left( {2x + 1} \right)\)

\(\Leftrightarrow (2x+1)(3x-2)-(5x-8)(2x+1)=0\)

\(\Leftrightarrow (2x+1)[(3x-2)-(5x-8)]=0\)

\(\Leftrightarrow (2x+1)(3x-2-5x+8)=0\)

\(\Leftrightarrow (2x+1)(-2x+6)=0\)

\(\Leftrightarrow \left[ \matrix{2x+1=0 \hfill \cr -2x+6=0 \hfill \cr} \right.\)

\(\Leftrightarrow \left[ \matrix{x=-\frac{1}{2} \hfill \cr x=3 \hfill \cr} \right.\)

Vậy phương trình có hai nghiệm là \(x=-\frac{1}{2}\)hoặc \(x=3\)

b)\(4{x^2} - 1 = \left( {2x + 1} \right)\left( {3x - 5} \right)\)

\(⇔( 2x - 1)( 2x + 1) = \left( {2x + 1} \right)\left( {3x - 5} \right)\)

\(⇔\left( {2x + 1} \right)\left( {2x - 1 - 3x + 5} \right)=0\)

\(⇔\left( {2x + 1} \right)\left( {4 - x} \right) = 0\)

\(⇔\left[ \matrix{2x+1=0 \hfill \cr 4-x=0 \hfill \cr} \right.\)

\(⇔\left[ \matrix{x=-\frac{1}{2} \hfill \cr x=4 \hfill \cr} \right.\)

Vậy phương trình có hai nghiệm \(x =-{1 \over 2}\)hoặc \(x = 4\)

c) \({\left( {x + 1} \right)^2} = 4\left( {{x^2} - 2x + 1} \right)\)

\(⇔(x+1)^2=4(x-1)^2\)

\(⇔(x+1)^2-4(x-1)^2=0\)

\(⇔(x+1)^2-(2x-2)^2=0\)

\(⇔(x+1-2x+2)(x+1+2x-2)=0\)

\(⇔(-x+3)(3x-1)=0\)

\(⇔\left[ \matrix{-x+3=0 \hfill \cr 3x-1=0 \hfill \cr} \right.\)

\(⇔\left[ \matrix{x=3 \hfill \cr x=\frac{1}{3} \hfill \cr} \right.\)

Vậy phương trình có hai nghiệm là \(x=3\)hoặc \(x=\frac{1}{3}\)

d) \(2{x^3} + 5{x^2} - 3x = 0\)

\(⇔x\left( {2{x^2} + 5x - 3} \right) = 0\)

\(⇔x(2x^2+6x-x-3)= 0\)

\(⇔x[(2x^2+6x)-(x+3)]= 0\)

\(⇔x\left[ {2x\left( {x + 3} \right) - \left( {x + 3} \right)} \right] = 0\)

\(⇔x\left( {x + 3} \right)\left( {2x - 1} \right) = 0\)

\(⇔\left[ {\matrix{{x = 0} \cr {x + 3 = 0} \cr {2x - 1 = 0} \cr} } \right. \)

\(\Leftrightarrow \left[ {\matrix{{x = 0} \cr {x = - 3} \cr {x = {1 \over 2}} \cr} } \right.\)

Vậy phương trình có ba nghiệm \(x = 0; x = -3; x ={1 \over 2}\)