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When it comes to polynomial subtraction, a critical first step is distributing the negative sign across the second polynomial. Think of the subtraction sign like a special set of invisible parentheses around the second polynomial. To "distribute" means to multiply each term in the second polynomial by -1. By turning each term into its opposite, you effectively change subtraction into addition.
Here’s how it works in practice:

• Originally, you have: \( (10y^2 - 7) - (20y^3 - 2y^2 - 7) \).
• Distribute the negative sign like this: transform \( - (20y^3 - 2y^2 - 7) \) into \( -20y^3 + 2y^2 + 7 \).

Now, the expression is: \( 10y^2 - 7 - 20y^3 + 2y^2 + 7 \). This allows you to proceed easily to the next step of combining like terms.

Once you have successfully distributed the negative sign, the next task is to find and combine like terms. "Like terms" refer to terms within an expression that have the exact same variable component.
In our expression, after distribution, you see terms like \(-20y^3\), \(10y^2\), \(2y^2\), and constants \(-7\) and \(+7\). Here's the breakdown:

• Combine \(10y^2\) and \(2y^2\) because they both have the same "\(y^2\)" variable, resulting in \(12y^2\).
• The terms \(-7\) and \(+7\) cancel each other out because they sum to zero.

With this, the expression simplifies to \(-20y^3 + 12y^2\). Once like terms are combined, all unrelated terms are left untouched, maintaining the integrity of the polynomial.

The final step in polynomial subtraction is simplifying the expression as much as possible. Simplification means expressing the polynomial in its simplest form, which often involves eliminating terms that cancel each other or combining all possible terms fully.
From our work above, we ended with \(-20y^3 + 12y^2 + 0\). The zero represents the cancelled-out constants, which we can now omit for clarity:

• The simplified polynomial becomes \(-20y^3 + 12y^2\).

No further steps are required since there are no more like terms to combine, and no coefficients to factor out.
At this point, you have a clean and simplified polynomial expression. Remember, simplifying doesn't change the value of the expression; it just expresses it in the most concise way possible.