How do you know which substitutions to make to cancel out a term?

How do you know which substitutions to make to cancel out a term?

I am doing problem B45 from Ivan Niven's "Maxima and Minima Without
Calculus" which says:
"Consider the quadratic polynomial $f(x, y)=ax^2+2bxy+cy^2+dx+cy+k$ ,
where the coefficients are real constants with $a>0$ and $c>0$ . In case
$b \not =0$ , verify that the transformation $x=X-by/a$ produces a
quadratic polynomial $f(X-by/a , y)$ with no $Xy$ term."
The reason why he didn't want an $Xy$ terms is because if there is none
then the function is easy to minimize by minimizing the two variables
separately. My question is, what steps led to the substitution $x=X-by/a$?
How did he know that making this substitution would lead you to having no
$Xy$ term? How can this method be extended more generally to other
functions? Thanks.