The problem is to expand the expression $(a + b + 5)^2$.

1. Problem Description

The problem is to expand the expression $(a + b + 5)^2$.

2. Solution Steps

We want to expand $(a + b + 5)^2$. We can think of this as $(a + b + 5)(a + b + 5)$.

We can use the multinomial theorem, or we can expand the expression by using the distributive property.

Method 1: Using the distributive property

$(a + b + 5)^2 = (a + b + 5)(a + b + 5)$

$= a(a + b + 5) + b(a + b + 5) + 5(a + b + 5)$

$= a^2 + ab + 5a + ba + b^2 + 5b + 5a + 5b + 25$

$= a^2 + b^2 + ab + ba + 5a + 5a + 5b + 5b + 25$

$= a^2 + b^2 + 2ab + 10a + 10b + 25$

Method 2: Using the formula $(x+y+z)^2 = x^2 + y^2 + z^2 + 2xy + 2xz + 2yz$

Let $x = a$, $y = b$, and $z = 5$. Then

$(a + b + 5)^2 = a^2 + b^2 + 5^2 + 2ab + 2(a)(5) + 2(b)(5)$

$= a^2 + b^2 + 25 + 2ab + 10a + 10b$

$= a^2 + b^2 + 2ab + 10a + 10b + 25$

3. Final Answer

$a^2 + b^2 + 2ab + 10a + 10b + 25$