Ta có: \(\left(2x+5\right)\left(4x^2-10x+25\right)-\left(2x+1\right)^2=52\)
<=> \(8x^3+125-4x^2-4x-1=52\)
<=> \(8x^3-4x^4-4x+72=0\)
<=> \(2x^3-x^2-x+18=0\)
<=> \(\left(2x^3+4x^2\right)-\left(5x^2+10x\right)+\left(9x+18\right)=0\)
<=> \(2x^2\left(x+2\right)-5x\left(x+2\right)+9\left(x+2\right)=0\)
<=> \(\left(x+2\right)\left(2x^2-5x+9\right)=0\)
<=> \(\left[{}\begin{matrix}x+2=0\left(1\right)\\2x^2-5x+9=0\left(2\right)\end{matrix}\right.\)
Mặt khác :
\(2x^2-5x+9=2\left(x^2-2,5x+1,5625\right)+5,875\)
= \(2\left(x-1,25\right)^2+5,875\) \(\ge5,875\) > 0
=> PT (2) vô nghiệm
=> x + 2 = 0 => \(x=-2\)
Vậy nghiệm của PT : S = \(\left\{-2\right\}\)
