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The process of common factor extraction is a key step in factoring polynomials. It involves identifying and removing a factor that is shared by all terms in a polynomial. To execute this, look for the greatest common factor (GCF), which includes numbers and variables that are present in each term.

For instance, the polynomial in our exercise was, \(5 y^{5}-5 y^{4}-20 y^{3}+20 y^{2}\). All terms share a \(5\) and a \(y^{2}\), making \(5 y^{2}\) the GCF. By extracting this factor, we simplify the complex polynomial into a more manageable form: \(5 y^{2}(y^{3} - y^{2} - 4y + 4)\). By respecting the distributive property of multiplication, we ensure that when the GCF is distributed back across the terms in parentheses, we indeed get our original polynomial. This step is vital for making the remaining factoring steps more straightforward.

A quadratic expression is a type of polynomial with a degree of 2, which means the highest power of the variable is 2. Its general form is \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a\) is non-zero.

The challenge within our exercise is that we are not dealing with a pure quadratic, but rather a cubic expression: \(y^{3} - y^{2} - 4y + 4\). However, by rearranging the terms into \((y^{3} - y^{2}) - 4(y - 1)\), it simplifies into two expressions, both of which contain a factor of \(y - 1\). Recognizing and rearranging such terms to reveal a hidden quadratic form is a crucial preliminary step before we apply the technique known as 'factor by grouping'.

The term factor by grouping refers to a technique where we group terms in a way that allows us to factor out common terms or expressions. This is particularly useful when a polynomial does not factor easily as a whole. After common factor extraction and recognizing quadratic forms, we are often left with an expression ripe for this strategy.

In our exercise, once we had \(y^{2}(y - 1) - 4(y - 1)\), we noticed that \(y - 1\) is common to both grouped terms. We can then factor out \(y - 1\), which gives us \(5y^{2}(y - 1)(y^{2} - 4)\) as the final factored form of the polynomial. It should be noted that the process of grouping and factoring can sometimes lead to further factorization opportunities, such as recognizing the difference of squares in the residual quadratic expression \(y^{2} - 4\).