To solve the given problem where [tex]\( a + 2 = a^2 + 2 \)[/tex], we need to simplify and solve for [tex]\( a \)[/tex]:
1. Start with the equation:
[tex]\[ a + 2 = a^2 + 2 \][/tex]
2. Subtract 2 from both sides to get rid of the constant term:
[tex]\[ a = a^2 \][/tex]
3. Rewrite it in the standard quadratic equation form:
[tex]\[ a^2 - a = 0 \][/tex]
4. Factor the quadratic equation:
[tex]\[ a(a - 1) = 0 \][/tex]
5. Set each factor to zero and solve for [tex]\( a \)[/tex]:
[tex]\[ a = 0 \quad \text{or} \quad a - 1 = 0 \quad \Rightarrow \quad a = 1 \][/tex]
Therefore, the solutions to the equation [tex]\( a + 2 = a^2 + 2 \)[/tex] are [tex]\( a = 0 \)[/tex] and [tex]\( a = 1 \)[/tex].
Now, we move to the calculation of [tex]\( 3 + 4 + 5 \)[/tex]:
Calculate the sum of the numbers:
[tex]\[ 3 + 4 + 5 = 12 \][/tex]
Thus, the result is:
[tex]\[ \boxed{12} \][/tex]
None of the given options (A) 2, (B) 0, (C) -1, (D) -2, (E) -3 match our result. Therefore, it seems there's an error in the provided options or an additional context might be missing. However, based on the calculation of the sum of the numbers:
The correct answer derived from the given problem should be [tex]\( 12 \)[/tex].
