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When factoring any polynomial, the first step is to look for a common factor. This is a number or variable that evenly divides all terms in the expression. Identifying a common factor simplifies the polynomial and makes further factoring easier.
In the given trinomial 50h^2 - 40hy + 8y^2, each term is divisible by 2. Factoring out the 2, we get:
\[ 50h^2 - 40hy + 8y^2 = 2(25h^2 - 20hy + 4y^2) \]
By simplifying the expression in this way, we reduce the complexity of the problem. The common factor is essential when you start working on polynomial factoring.

A trinomial is a polynomial with exactly three terms. In this problem, the trinomial is 25h^2 - 20hy + 4y^2. Trinomials can often be factored into the product of two binomials.

To factor the given trinomial, we look for binomials that multiply to produce the original expression. In this case, observe that the given trinomial resembles a perfect square trinomial:
\[ a^2 - 2ab + b^2 = (a - b)^2 \]
Given 25h^2 - 20hy + 4y^2, 25h^2 can be written as (5h)^2 and 4y^2 as (2y)^2. The middle term -20hy matches -2ab, where a is 5h and b is 2y. Hence, we can rewrite and factor the trinomial as:
\[ 25h^2 - 20hy + 4y^2 = (5h - 2y)(5h - 2y) = (5h - 2y)^2 \]

A binomial is a polynomial with exactly two terms. When factoring trinomials, we often break them down into the product of two binomials. Once we factor out the common factor from the trinomial, our goal is to find pairs of binomials that multiply to produce the original trinomial.

For example, starting with 25h^2 - 20hy + 4y^2, we rewrite it by factoring it as a square of a binomial:
\[ (5h - 2y)(5h - 2y) \]
This means the trinomial can be expressed as the square of the binomial (5h - 2y):
\[ (5h - 2y)^2 \]
Finally, combining it back with the common factor we factored out initially, we get:
\[ 2(5h - 2y)(5h - 2y) \]
Factoring polynomials like this helps in understanding deeper algebraic concepts and simplifies solving higher-degree polynomial equations.