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Using both continuous and categorical/indicator variables in a linear regression model is perfectly fine. For example, you can look at this post that describes several methods to code categorical variables for regression analyses, or this post. However, you should avoid the dummy variable trap, where several dummy variables are correlated to each other.
If ...
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We need to take some care with the notation because the models differ.
Let the first (correct) model be
$$Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \varepsilon\tag{1}$$
where the $\varepsilon_i$ have a common variance and zero means; and write the second model (which governs the very same variables $Y$, so no need to change their name) as
$$Y = \alpha_0 +...
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A reasonable approach for your problem is forward selection, a type of Stepwise Regression.
You could fit a regression with just the $A$ variables, and then compare this to the regression with the $A,B$, $A,C$, and $A,D$ variables via F-test where, for example
$$F_{A \; \text{vs.} \, A,B}=\frac{(SSE_A-SSE_{A,B})/(df_{A}-df_{A,B})}{SSE_{A,B}/df_{A,B}}.$$
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