1. The problem is to simplify the expression $$(A + \beta + C) i \overline{\beta}$$ where $A$, $\beta$, and $C$ are variables, $i$ is the imaginary unit, and $\overline{\beta}$ is the complex conjugate of $\beta$. 2. Recall that the complex conjugate $\overline{\beta}$ of a complex number $\beta = x + yi$ is $x - yi$. 3. The expression involves multiplication of a sum by $i \overline{\beta}$, so we distribute: $$ (A + \beta + C) i \overline{\beta} = A i \overline{\beta} + \beta i \overline{\beta} + C i \overline{\beta} $$ 4. Each term is a product of constants/variables with $i$ and $\overline{\beta}$. Without specific values or further relations, this is the simplified expanded form. 5. If $\beta$ is a complex number, then $\beta \overline{\beta} = |\beta|^2$ (the squared magnitude), so the middle term can be rewritten: $$ \beta i \overline{\beta} = i |\beta|^2 $$ 6. Thus, the expression becomes: $$ A i \overline{\beta} + i |\beta|^2 + C i \overline{\beta} = i \overline{\beta} (A + C) + i |\beta|^2 $$ 7. This is the simplified form showing the terms clearly. Final answer: $$ (A + \beta + C) i \overline{\beta} = i \overline{\beta} (A + C) + i |\beta|^2 $$