Expected

Output

In[]:=
expectedOutput={3+λ1+3λ2,2-2λ1-λ2,-7-λ1+2λ2,-15-13λ1-24λ2,5+11λ1+8λ2,40λ1};

Solution

Equation

In[]:=
eq=40λ1+x(-15-13λ1-24λ2)+y^2(2-2λ1-λ2)+xy(-7-λ1+2λ2)+x^2(3+λ1+3λ2)+y(5+11λ1+8λ2)==0;
In[]:=
cl=CoefficientList[eq[[1]],{x,y}];​​coefficients=Flatten[​​{​​cl[[3,1]],​​cl[[1,3]],​​cl[[2,2]],​​cl[[2,1]],​​cl[[1,2]],​​cl[[1,1]]​​}​​]
Out[]=
{3+λ1+3λ2,2-2λ1-λ2,-7-λ1+2λ2,-15-13λ1-24λ2,5+11λ1+8λ2,40λ1}

Compare

In[]:=
coefficients==expectedOutput
Out[]=
True