Let's simplify the given expression \( y = (x + 1)^2 + (x + 2)^2 \).

1. Expand the squares:
   \[
   (x + 1)^2 = x^2 + 2x + 1
   \]
   \[
   (x + 2)^2 = x^2 + 4x + 4
   \]

2. Substitute these into the equation for \( y \):
   \[
   y = (x^2 + 2x + 1) + (x^2 + 4x + 4)
   \]

3. Combine like terms:
   \[
   y = x^2 + x^2 + 2x + 4x + 1 + 4
   \]
   \[
   y = 2x^2 + 6x + 5
   \]

So, the simplified form of the expression is:
\[
y = 2x^2 + 6x + 5
\]

Would you like more details or have any questions? Here are some related questions:

1. How do you factor quadratic expressions like \(2x^2 + 6x + 5\)?
2. What is the vertex form of the quadratic equation \(y = 2x^2 + 6x + 5\)?
3. How can you find the roots of the quadratic equation \(2x^2 + 6x + 5 = 0\)?
4. What is the significance of the discriminant in a quadratic equation?
5. How do you complete the square for the expression \(2x^2 + 6x + 5\)?

**Tip:** Always check if a quadratic expression can be factored before applying the quadratic formula.

Learn how to simplify the quadratic expression (x + 1)^2 + (x + 2)^2 with detailed step-by-step solutions. This problem covers key algebraic concepts, including expanding squares and combining like terms. Suitable for advanced high school students.

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