https://file.aiquickdraw.com/mathbot/file/1727963430670reoy13gx.jpg

The expression in the image is:

\[
(x + y)\left[(x + z)(y + z) - z^2\right] - x(xy + zx - yz) + y(xz - yx - yz)
\]

Let’s simplify it step-by-step:

### Step 1: Expand the terms inside the first parenthesis
We have \((x + z)(y + z)\), which expands as:

\[
(x + z)(y + z) = xy + xz + yz + z^2
\]

Thus, the expression inside the first parenthesis becomes:

\[
(x + z)(y + z) - z^2 = xy + xz + yz + z^2 - z^2 = xy + xz + yz
\]

### Step 2: Substitute and simplify the first part
Substitute \(xy + xz + yz\) into \((x + y)[(x + z)(y + z) - z^2]\):

\[
(x + y)(xy + xz + yz) = x(xy + xz + yz) + y(xy + xz + yz)
\]

Expanding:

\[
x(xy + xz + yz) = x^2y + x^2z + xyz
\]
\[
y(xy + xz + yz) = yx^2 + yxz + y^2z
\]

Thus:

\[
(x + y)(xy + xz + yz) = x^2y + x^2z + xyz + yx^2 + yxz + y^2z
\]

### Step 3: Simplify the second and third terms
The second term is:

\[
-x(xy + zx - yz) = -x^2y - x^2z + xyz
\]

The third term is:

\[
y(xz - yx - yz) = yxz - y^2x - y^2z
\]

### Step 4: Combine all terms
Putting it all together:

\[
(x^2y + x^2z + xyz + yx^2 + yxz + y^2z) - (x^2y + x^2z - xyz) + (yxz - y^2x - y^2z)
\]

This results in:

\[
yx^2 + y^2x + 2yxz
\]

### Final simplified expression
The final simplified form of the expression is:

\[
yx^2 + y^2x + 2yxz
\]

Would you like me to expand on this explanation or verify any specific step?

### Additional Questions
1. Would you like to see this process broken down further?
2. Are there any similar algebraic expressions you want to practice with?
3. Should I substitute specific values to show a concrete example?
4. Do you need help visualizing these expansions?
5. Are you interested in factoring the final simplified expression?

### Tip
When simplifying complex expressions, try breaking down each section step-by-step and look for patterns in terms that may cancel or combine.

Simplify the given algebraic expression: (x + y)[(x + z)(y + z) - z^2] - x(xy + zx - yz) + y(xz - yx - yz).

Learn how to simplify a complex algebraic expression involving three variables (x, y, z). This solution covers expanding expressions, applying the distributive property, and combining like terms. Suitable for students in Grades 10-12.

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