Laura teacher took a class test on factorisation she wrote the following sum on the class board factorise (x-y)square+4xy-zsquare step 1(xsquare-2xy+ysquare)+4xy-zsquare step 2(x square -2xy+y square)-z square step 3 (x-y)square-zsquare step 4(x-y+z)(x-y-z) what is the correct step she should write in the place of wrong step 

To correct the mistake Laura made in Step 2, let's go through the problem step by step.

### Given expression:
\[
(x - y)^2 + 4xy - z^2
\]

### Correct Step 1 (Expanding \((x - y)^2\)):
\[
x^2 - 2xy + y^2 + 4xy - z^2
\]
This is correct so far.

### Correct Step 2 (Combining like terms):
Combine \(-2xy\) and \(4xy\) to get \(2xy\):
\[
x^2 + 2xy + y^2 - z^2
\]
This should be the correct Step 2.

### Step 3 (Recognizing a perfect square and difference of squares):
Recognize that \(x^2 + 2xy + y^2\) is a perfect square:
\[
(x + y)^2 - z^2
\]

### Step 4 (Factorizing using the difference of squares formula):
\[
(x + y + z)(x + y - z)
\]

### Summary:
The correct Step 2 should be:
\[
x^2 + 2xy + y^2 - z^2
\]

Would you like more details on these steps? Here are some related questions:

1. How can you identify perfect square trinomials?
2. What is the difference of squares, and how do you factor it?
3. Why is it important to combine like terms before proceeding with factorization?
4. Can you explain the process of recognizing and factoring a perfect square trinomial?
5. What other common patterns are useful for factorization in algebra?
6. How can errors in early steps affect the final solution in algebraic problems?
7. What strategies can help avoid common mistakes in factorization?
8. How would the factorization process change for a different quadratic expression?

**Tip:** Always combine like terms before attempting to factorize; this simplifies the expression and makes patterns easier to spot.

Find the correct step in factorizing the expression (x-y)^2 + 4xy - z^2 with a detailed solution. Learn about perfect square trinomials, the difference of squares formula, and algebraic simplification techniques. Ideal for high school students.

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