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Understanding the concept of coefficient identification is the first crucial step in factoring quadratic trinomials. Coefficients are simply the numbers that multiply the variables in an algebraic expression. In the quadratic trinomial form \( ax^2 + bx + c \), \( a \) is the coefficient of \( x^2 \) (the quadratic term), \( b \) is the coefficient of \( x \) (the linear term), and \( c \) is the constant term. In our example \( 2y^2 - 4y + 2 \), the coefficient identification reveals that \( a=2 \) for the \( y^2 \) term, \( b=-4 \) for the \( y \) term, and \( c=2 \) for the constant.
To prepare for factoring, it's important to correctly identify these coefficients as they directly influence the search for factors that will simplify the quadratic expression into something more manageable.

Once the coefficient identification has been tackled, the next step is searching for factors of quadratic equations. This means finding two numbers that can fill two critical roles simultaneously: when added together, they produce the middle term's coefficient, \( b \), and when multiplied, they yield the product of the quadratic coefficient \( a \) and the constant \( c \). For the specified exercise, our target numbers need to fulfill the equation \( n + m = b \) and \( n * m = a * c \) simultaneously. It turns out that \( -2 \) and \( -2 \) are the magic numbers, as \( -2 - 2 = -4 \) (matching \( b \) from our trinomial) and \( -2 * -2 = 4 \) (the product of \( a \) and \( c \) in our case). This search step is essential for correctly rewriting the original quadratic in a form that will reveal its factors.

With the factors in hand, the process to rewrite and factor quadratic expressions becomes straightforward. This involves transforming the original quadratic trinomial into an equivalent expression where the factors are apparent. For our exercise, the expression \( 2y^2 - 4y + 2 \) can be rewritten by incorporating the numbers found in the previous step. It is vital to ensure that the new expression represents the same relationship as these factors come into play. Therefore, \( 2y^2 - 4y + 2 \) is rewritten as \( 2(y^2 - 2y + 1) \) by grouping the terms and taking out the common factor of 2, followed by recognizing the perfect square trinomial \( (y - 1)^2 \) within the parentheses. As a result, the factored form is \( 2(y - 1)^2 \) making it evident how \( (y - 1) \) as a factor is squared and scaled by the coefficient 2. This final form is not only simpler but also highly beneficial for further algebraic manipulation.