Simplify and Expand: The Expression (x+y+1)²

To solve this problem, we'll simplify the expression $(x+y+1)^2$ by recognizing it as a square of a sum involving three terms:

• Step 1: Use the formula $(a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc$.
• Step 2: Identify $a = x$, $b = y$, and $c = 1$.
• Step 3: Substitute these values into the formula.
• Step 4: Calculate each square and product term.
• Step 5: Simplify the expression by combining all computed terms.

Now, let's work through the steps:

We start with the formula: $(x+y+1)^2 = x^2 + y^2 + 1^2 + 2xy + 2 \cdot x \cdot 1 + 2 \cdot y \cdot 1$

Calculate each component:

• $x^2$ remains $x^2$.
• $y^2$ remains $y^2$.
• $1^2$ results in $1$.
• The term $2xy$ results from the cross-product of $x$ and $y$.
• The term $2 \cdot x \cdot 1$ simplifies to $2x$.
• The term $2 \cdot y \cdot 1$ simplifies to $2y$.

Combine these elements to form the simplified expression:

$x^2 + y^2 + 1 + 2xy + 2x + 2y$

Thus, the simplified expression for $(x+y+1)^2$ is:

$x^2 + 2x + y^2 + 2y + 2xy + 1$.

This corresponds to choice number 4 in the provided options.