Related Roots

Suppose the equation  
\[z^2 +2z+5=0\]
  has roots  
\[\alpha , \; \beta\]
.
For the equation above  
\[\alpha + \beta =-2, \; \alpha \beta =5\]
.
What are the values of  
\[(2 \alpha +1)(2 \beta +1), \; (2 \alpha +1)+(2 \beta +1) \]

\[(2 \alpha +1), \; (2 \beta +1)\]
  are called related roots.
\[(2 \alpha +1)(2 \beta +1)=4 \alpha \beta + 2( \alpha + \beta )+1 =4(5)+2(-2)+1=17\]

\[(2 \alpha +1)+(2 \beta +1)= 2( \alpha + \beta )+2 =2(-2)+2=-2\]
.
The quadratic equation with roots equal to these related roots is  
\[z^2+2z+17=0\]
.
Roots are related by a linear transformation. For it to be possible to write down an equation in this way, roots must be related by the same transformation.

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