$\text{a)}{x}^2\left(x+1\right)+x\left(x+1\right)+7\left(x+1\right)=0$  

$\left(x+1\right)\left(x^2+x+7\right)=0$  

$x+1=0\text{lub}\underset{\Delta =1-28<0}{x^2+x+7=0}$  

$x=-1$  

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$\text{b)}{x}^2\left(2x-5\right)-x\left(2x-5\right)+6\left(2x-5\right)=0$  

$\left(2x-5\right)\left(x^2-x+6\right)=0$  

$2x-5=0\text{lub}\underset{\Delta =1-24<0}{x^2-x+6}=0$  

$2x=5\mid :2$  

$x=\frac{5}{2}$  

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$\text{c)}{x}^2\left(x^2+1\right)+3x\left(x^2+1\right)+6\left(x^2+1\right)=0$  

$\left(x^2+1\right)\left(x^2+3x+6\right)=0$  

$x^2+1=0\text{lub}\underset{\Delta =9-24<0}{x^2+3x+6=0}$  

$\underset{\text{sprzecznosˊcˊ}}{x^2=-1}$  

Równanie nie ma rozwiązań.

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$\text{d)}{x}^2\left(3x-1\right)-4x\left(3x-1\right)+4\left(3x-1\right)=0$  

$\left(3x-1\right)\left(x^2-4x+4\right)=0$  

$\left(3x-1\right){\left(x-2\right)}^2=0$  

$3x-1=0\text{lub}{\left(x-2\right)}^2=0$  

$3x=1\mid :3\text{lub}x-2=0$  

$x=\frac{1}{3}\text{lub}x=2$  

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$\text{e)}{x}^2\left(2x-7\right)+6x\left(2x-7\right)+9\left(2x-7\right)=0$  

$\left(2x-7\right)\left(x^2+6x+9\right)=0$  

$\left(2x-7\right){\left(x+3\right)}^2=0$  

$2x-7=0\text{lub}{\left(x+3\right)}^2=0$  

$2x=7\mid :2\text{lub}x+3=0$  

$x=\frac{7}{2}\text{lub}x=-3$  

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$\text{f)}{x}^2\left(x^2+x\right)-x\left(x^2+x\right)+\left(x^2+x\right)=0$  

$\left(x^2+x\right)\left(x^2-x+1\right)=0$  

$x^2+x=0\text{lub}\underset{\Delta =1-4<0}{x^2-x+1=0}$  

$x\left(x+1\right)=0$  

$x=0\text{lub}x+1=0$  

$x=0\text{lub}x=-1$