To factorize the expression \( 2x^2 - k - 3 + 2y^2 - y + 4xy \), we will rearrange and group terms that can be factored more easily.

1. Start by writing the expression:
   \[
   2x^2 + 2y^2 + 4xy - y - k - 3
   \]

2. Group terms with similar variables:
   \[
   (2x^2 + 2y^2 + 4xy) - y - (k + 3)
   \]

3. Factor the quadratic terms related to \(x\) and \(y\):
   \[
   2(x^2 + y^2 + 2xy) - y - (k + 3)
   \]
   Notice that \(x^2 + y^2 + 2xy\) is a perfect square trinomial, which can be factored as \((x + y)^2\):
   \[
   2(x + y)^2 - y - (k + 3)
   \]

4. Now we have the expression:
   \[
   2(x + y)^2 - y - (k + 3)
   \]

This is the factorized form, but the terms involving \(y\) and the constants cannot be simplified further unless more information about \(k\) is provided.

Would you like a detailed explanation of any step?

Here are 5 related questions for further practice:

1. How do you factorize a quadratic expression with two variables?
2. What is the process of recognizing a perfect square trinomial?
3. Can you simplify \(2(x + y)^2 - y - (k + 3)\) further if \(k\) is a specific value?
4. How do you factor expressions involving constants and variables separately?
5. What is the next step if you were to expand the factorized expression?

**Tip**: Always check for common patterns, such as perfect squares or common factors, to simplify your factorization process.

Learn how to factorize the algebraic expression 2x² - k - 3 + 2y² - y + 4xy step-by-step. This problem covers key concepts in algebra such as factoring quadratic expressions and recognizing perfect square trinomials. Suitable for students in Grades 9-12.

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