To solve the differential equation given by

$\frac{dy}{dx} = \frac{4x + 3y + 2}{3x + 2y + 1},$

you can use methods for first-order differential equations. However, the equation is not separable nor does it fit a standard exact form or an easily recognizable linear or Bernoulli form, which makes it necessary to consider other methods such as substitutions or numerical solutions, depending on the context.

Here, a strategy for simplifying or solving such an equation would be to attempt some form of substitution to simplify the expression, or rearrange terms if there's an apparent opportunity for standardization. Often, transformations such as linear substitutions of the form $y = vx$ or attempting to express the equation in a form indicative of a homogenous function might be useful—though this equation does not readily suggest an obvious substitution.

If the goal is a general solution and no specific technique is noted or simplification is obvious, you may ultimately resort to numerical methods for approximate solutions if specific initial conditions are provided.

If you're dealing with a particular problem context that suggests a particular method (such as a boundary or initial condition), that additional information might guide a choice of technique. If that is the case, or if there are specific aspects you are interested in, feel free to provide further context!