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Understanding perfect square trinomials is key to factoring certain types of polynomials easily. A perfect square trinomial is a special form of polynomial that can be written as the square of a binomial. Generally, it looks like this: \(a^2 + 2ab + b^2\). Here, both \(a\) and \(b\) are terms, and the middle term is twice the product of these two terms. For example, in the expression \(x^2 + y^2 + 2xy\), we can identify this as a perfect square trinomial where \(a = x\) and \(b = y\). This identification helps us quickly rewrite the expression as \((x + y)^2\). Recognizing such patterns will simplify your factoring process considerably.

Another useful technique in polynomial factoring is to first factor out any common terms. This simplifies the polynomial, making it easier to work with. In the given problem, \(-x^2 - y^2 - 2xy\), note that every term has a leading negative sign. Factoring out \(-1\) simplifies the expression from \(-x^2 - y^2 - 2xy\) to \(-1(x^2 + y^2 + 2xy)\). By doing this, we convert our problem into one involving a positive perfect square trinomial, which is easier to factor.

Simplifying expressions is the final step in making your factored polynomials as neat and understandable as possible. Once you have factored out any common terms and identified any perfect square trinomials, your next job is to combine and simplify these parts. For the example in our solution, after recognizing \((x^2 + y^2 + 2xy)\) as a perfect square trinomial, we rewrote it as \((x + y)^2\). We then combine it with the \(-1\) we factored out initially, giving us \(-1(x + y)^2\). This simplified result is written as \(-(x + y)^2\), which is the cleanest representation of the factored original polynomial. Simplifying your factored form into a concise expression is always the goal in polynomial factoring.