Pedagogically speaking, factoring is a lot less intuitive than 'simple' rearrangement. For your example we have that, $$ 2x +4 =10. $$ When first teaching Algebra, there are many nice and neat tricks/visualizations to understand the process of unraveling the equation to solve for $x$. A classic analogy is to see the equation as a kind of seesaw that's balanced and you need to do steps so that the see-saw is always balanced.

A cute way that I was taught was to role play as a greedy family lawyer who had to 'divorce' $x$ from it's current relationship with the numbers it is with by doing actions that oppose what holds their relationship together in the first place.

So there are a lot of ways to explain this to someone who's first learning about something pretty abstract. In comparison, to get the solution by factoring does not have any nice analogy that can be used. In essence, we need to ask: what value of $x$ must be satisfied such that the RHS is zero? Which to an untrained mind is an extra layer of abstraction that doesn't need to be added until they're already comfortable with manipulating equations.

Further, it's even harder with the factoring method because the solution splits off into two 'branches'. Which is once again not immediately obvious to students why that should be the case. This problem can be swept under the rug with $\pm$.

However, I guess there are things that need to be untrained with the method of straight manipulation. So I agree that a greater class of problems can be solved a lot more straightforwardly by factoring, especially when working with $\mathbb{C}$.

Regardless, I think that usually people have a hard time learning Algebra for the first time and so we need to make this initial step up into abstraction as easy as possible.

Pedagogically speaking, factoring is a lot less intuitive than 'simple' rearrangement. For your example we have that, $$ 2x +4 =10. $$ When first teaching Algebra, there are many nice and neat tricks/visualizations to understand the process of unraveling the equation to solve for $x$. A classic analogy is to see the equation as a kind of seesaw that's balanced and you need to do steps so that the see-saw is always balanced.

A (perhaps insensitive) way that I was taught was to role play as a greedy family lawyer who had to 'divorce' $x$ from it's current relationship with the numbers it is with by doing actions that oppose what holds their relationship together in the first place.

So there are a lot of ways to explain this to someone who's first learning about something pretty abstract. In comparison, to get the solution by factoring does not have any nice analogy that can be used. In essence, we need to ask: what value of $x$ must be satisfied such that the RHS is zero? Which to an untrained mind is an extra layer of abstraction that doesn't need to be added until they're already comfortable with manipulating equations.

Further, it's even harder with the factoring method because the solution splits off into two 'branches'. Which is once again not immediately obvious to students why that should be the case. This problem can be swept under the rug with $\pm$.

However, I guess there are things that need to be untrained with the method of straight manipulation. So I agree that a greater class of problems can be solved a lot more straightforwardly by factoring, especially when working with $\mathbb{C}$.

Regardless, I think that usually people have a hard time learning Algebra for the first time and so we need to make this initial step up into abstraction as easy as possible.