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In algebra, distributing negative signs is a crucial step. It's called the 'distributive property,' which means you apply the negative sign to each term inside the parentheses. For instance, in the given expression \( x^{2} - (x^{2} - y^{2} - z) \), you need to distribute the negative sign to \(x^{2} - y^{2} - z\).
This changes every term inside to its opposite. The terms inside the parentheses will be affected as follows:

• \- x^{2}\ becomes \+ x^{2}\ (since the double negative turns positive)
• \-y^{2}\ becomes \+y^{2}\
• \-z\ becomes \+z\

As a result, the expression simplifies to \[ x^2 - x^2 + y^2 + z \]. This alteration helps get rid of the parentheses and prepares us for the next step. Distributing negative signs correctly is vital for the simplification process because one wrong sign can throw off the entire expression.

Once the expression has no more parentheses, the next step is to combine like terms. Like terms have the same variable raised to the same power. For example, \(x^{2}\) and \(-x^{2}\) are like terms because they both contain \(x^{2}\).
In our expression, after distributing the negative sign, we have: \[ x^2 - x^2 + y^2 + z \]
Combining \(x^{2}\) and \(- x^{2}\) gives: \[ x^2 - x^2 = 0 \]
This eliminates the \(x^{2}\) terms, simplifying the expression to: \[ 0 + y^2 + z \].
Remember, combining like terms is about eliminating redundancy and simplifying the expression. Look for terms with the same variable and powers to combine them effectively. This step makes the rest of the process more straightforward.

Simplification is the final step where all pieces of the expression come together in a more concise form. After combining like terms, we are left with: \[ 0 + y^2 + z \].
Combining the zero with other terms does not change their values: \[ 0 + y^2 + z = y^2 + z \].
So, the simplified form of the given expression is: \[ y^2 + z \].
Simplification is about making expressions easier to understand and use. It's essential in solving equations, making computations more accessible, and unveiling clearer insights within mathematical problems.
Remember, the aim is to reduce the expression to its simplest form while retaining its original meaning. By following through these steps, you ensure accuracy and clarity in your work.