To solve the given mathematical expression step-by-step, let’s break it down and simplify it.
The expression given is:
[tex]\[
R = (a + 5)^2 + (a + 6)^2 - 2(a^2 + 11a)
\][/tex]
Let's start by expanding each term individually:
1. Expand [tex]\((a + 5)^2\)[/tex]:
[tex]\[
(a + 5)^2 = a^2 + 10a + 25
\][/tex]
2. Expand [tex]\((a + 6)^2\)[/tex]:
[tex]\[
(a + 6)^2 = a^2 + 12a + 36
\][/tex]
3. Expand [tex]\(2(a^2 + 11a)\)[/tex]:
[tex]\[
2(a^2 + 11a) = 2a^2 + 22a
\][/tex]
Substitute these expanded forms back into the original expression:
[tex]\[
R = (a^2 + 10a + 25) + (a^2 + 12a + 36) - 2(a^2 + 11a)
\][/tex]
Combine like terms:
[tex]\[
R = a^2 + 10a + 25 + a^2 + 12a + 36 - 2a^2 - 22a
\][/tex]
Simplify by combining all the [tex]\(a^2\)[/tex], [tex]\(a\)[/tex], and constant terms:
[tex]\[
R = (a^2 + a^2 - 2a^2) + (10a + 12a - 22a) + (25 + 36)
\][/tex]
[tex]\[
R = 0a^2 + 0a + 61
\][/tex]
[tex]\[
R = 61
\][/tex]
Thus, the simplified expression yields:
[tex]\[
R = 61
\][/tex]
Therefore, the answer is:
[tex]\[
\boxed{61}
\][/tex]
