How to keep the specified term unchanged and move other terms to the right of the equation?

Questions
1

The polynomial is:

x2 y0 (-b^2 - a^2 \[Lambda]) + 
  x0 y2 (-b^2 - a^2 \[Lambda]) + (x2 y1 - x1 y2) (b^2 - 
     a^2 \[Lambda]) + x1 y0 (b^2 + a^2 \[Lambda]) + 
  x0 y1 (b^2 + a^2 \[Lambda]) == 0

Keep the specified items (x2 y1- x1 y2) (b ^ 2- a ^ 2 [Lambda]) unchanged and move the other items to the right of the equation. The desired result is as follows:

(x2 y1 - x1 y2) (b^2 - 
    a^2 \[Lambda]) == -x2 y0 (-b^2 - a^2 \[Lambda]) - 
  x0 y2 (-b^2 - a^2 \[Lambda]) - x1 y0 (b^2 + a^2 \[Lambda]) - 
  x0 y1 (b^2 + a^2 \[Lambda])

But I tried using the following code operation but didn’t get the desired answer. How can I correct the code?

x2 y0 (-b^2 - a^2 \[Lambda]) + 
  x0 y2 (-b^2 - a^2 \[Lambda]) + (x2 y1 - x1 y2) (b^2 - 
     a^2 \[Lambda]) + x1 y0 (b^2 + a^2 \[Lambda]) + 
  x0 y1 (b^2 + a^2 \[Lambda]) == 0
% /. k_.*((x2y1_) - (x1y2_)) + t_ == 0 :> k (x2y1 - x1y2) == -t

get the error answer:

x2 y0 (-b^2 - a^2 \[Lambda]) + 
  x0 y2 (-b^2 - a^2 \[Lambda]) + (x2 y1 - x1 y2) (b^2 - 
     a^2 \[Lambda]) + x1 y0 (b^2 + a^2 \[Lambda]) + 
  x0 y1 (b^2 + a^2 \[Lambda]) == 0
2

Does this do exactly what you want for this specific example?

poly=x2 y0(-b^2-a^2 λ)+x0 y2(-b^2-a^2 λ)+(x2 y1-x1 y2)(b^2-a^2 λ)+x1 y0(b^2+a^2 λ)+x0 y1(b^2+a^2 λ);
term=(x2 y1-x1 y2)(b^2-a^2 λ);
term==term-poly

which returns

(x2 y1-x1 y2)(b^2-a^2 λ)== -(x2 y0(-b^2-a^2 λ))-x0 y2(-b^2-a^2 λ)-x1 y0(b^2+a^2 λ)-x0 y1(b^2+a^2 λ)

Trying to do this is perhaps risky and trying to fight Mathematica at times. It may want to format things the way it wants and you want to format things the way you want. If you touch that result in some way it is possible that it will be reformatted in a way that you don’t want.

3

. k.*((x2y1) - (x1y2)) + t == 0 :> k (x2y1 - x1y2) == -t

How to use pattern matching methods?