The problem asks to expand the expression $(a+b+1)(a+b+2)$.

1. Problem Description

The problem asks to expand the expression $(a+b+1)(a+b+2)$.

2. Solution Steps

Let $x = a+b$. Then the expression becomes $(x+1)(x+2)$.

Expanding this expression, we get:

$(x+1)(x+2) = x(x+2) + 1(x+2)$

$= x^2 + 2x + x + 2$

$= x^2 + 3x + 2$

Substituting $x = a+b$ back into the expression, we get:

$(a+b)^2 + 3(a+b) + 2$

Expanding $(a+b)^2$, we use the formula:

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

Substituting this back into the expression, we get:

$a^2 + 2ab + b^2 + 3(a+b) + 2$

$= a^2 + 2ab + b^2 + 3a + 3b + 2$

3. Final Answer

$a^2 + b^2 + 2ab + 3a + 3b + 2$