## Subject III. (20 points)

Let $E_{1}=\frac{a(2 a+8)+8}{(a+4)(2 a+12)-16}$ and $E_{2}=\frac{a(a+4)+4}{a(a+1)-2}$.

a) Determine $a \in \mathbb{Z}$ such that the expressions make sense;

b) Determine $a \in \mathbb{Z}$ for which $\frac{1}{E_{1}}+\frac{1}{E_{2}} \in \mathbb{N}$;

c) Calculate $a \in \mathbb{Z}$ such that $\frac{1}{E_{1}}+\frac{1}{E_{2}}=\frac{3 a+10}{a+2}$.

Prof. Simona Pop, Colegiul Augustin Maior Cluj-Napoca
## Subject III.

a) $(a+4)(2 a+12)-16=2(a+2)(a+8) ; a(a+1)-2=(a-1)(a+2) \quad \Rightarrow a \in Z \backslash\{-8,-2,1\}$

b) $\frac{1}{E_{1}}+\frac{1}{E_{2}}=\frac{2 a+7}{a+2} \in N \Rightarrow \quad a \in\{-1,-5,1\}$

c) $a=-3$