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.