$a)$  

$8x-2x\left(3x-1\right)=2x^2+3$  

$8x-6x^2+2x=2x^2+3$  

$8x^2-10x+3=0$  

$\Delta =100-96=4$  

$\sqrt{\Delta }=2$  

 

$x_1=\frac{10+2}{16}=\frac{3}{4}$   

$x_2=\frac{10-2}{16}=\frac{1}{2}$  

 

$x\in \left\{\frac{3}{4},\frac{1}{2}\right\}$  

 

$b)$      

$x^2-5x+9=63-\left(x-3\right)\left(x+3\right)$  

$x^2-5x+9=63-x^2+9$  

$2x^2-5x-63=0$  

$\Delta =25+504=529$  

$\sqrt{\Delta }=23$  

 

$x_1=\frac{5-23}{4}=-\frac{9}{2}$  

$x_2=\frac{5+23}{4}=7$  

 

$x\in \left\{-\frac{9}{2},7\right\}$  

 

$c)$  

$\left(2x-4\right)\left(2x+4\right)={\left(3x-1\right)}^2-16$  

$4x^2-16=9x^2-6x+1-16$  

$5x^2-6x+1=0$  

$\Delta =36-20=16$  

$\sqrt{\Delta }=4$  

 

$x_1=\frac{6-4}{10}=\frac{2}{10}=\frac{1}{5}$  

$x_2=\frac{6+4}{10}=1$  

 

$x\in \left\{\frac{1}{5},1\right\}$  

 

$d)$  

$6-{\left(1-x\right)}^2={\left(2x-3\right)}^2-{\left(3x+2\right)}^2$  

$6-1+2x-x^2=4x^2-12x+9-9x^2-12x-4$  

$-4x^2-26x=0$  

$-2x\left(2x+13\right)=0$   

$x=0\vee x=-\frac{13}{2}$  

 

$x\in \left\{0,-\frac{13}{2}\right\}$  

 

$e)$  

$\left(x^2-3x+1\right)\left(x+2\right)=x^3-2x-1$  

$x^3-3x^2+x+2x^2-6x+2=x^3-2x-1$  

$-x^2-3x+3=0$  

$\Delta =9+12=21$  

$\sqrt{\Delta }=\sqrt{21}$  

 

$x_1=\frac{3-\sqrt{21}}{-2}$  

$x_2=\frac{3+\sqrt{21}}{-2}$  

 

$x\in \left\{\frac{3-\sqrt{21}}{-2},\frac{3+\sqrt{21}}{-2}\right\}$  

 

$f)$  

$\left(x-5\right)\left(x^2+5x+25\right)=x\left(x^2-4x\right)-124$  

$x^3+5x^2+25x-5x^2-25x-125=x^3-4x^2-124$  

$-4x^2+1=0$  

$x^2=\frac{1}{4}$  

$x=\frac{1}{2}\vee x=-\frac{1}{2}$  

 

$x\in \left\{-\frac{1}{2},\frac{1}{2}\right\}$