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A perfect square trinomial is a specific type of quadratic expression. It takes the form \[a^2 ± 2ab + b^2\]. To recognize this pattern, check if the first and last terms are perfect squares and if the middle term equals twice the product of the square roots of the first and last terms.

In the original exercise, \(49x^2 - 28xy + 4y^2\), both \(49x^2\) and \(4y^2\) are perfect squares. To identify if it's a perfect square trinomial, check that \(49x^2\) can be written as \((7x)^2\) and \(4y^2\) as \((2y)^2\).

The middle term, \(-28xy\), should be equal to \(-2 \times 7x \times 2y\). Here, indeed \(-2 \times 7x \times 2y = -28xy\), confirming it fits the pattern of a perfect square trinomial.

Binomial expansion is the process of multiplying a binomial by itself multiple times. Essentially, if you have a binomial \((a \text{±} b)\) and raise it to a power, you expand it to multiple terms using specific coefficients.

For example, \((a + b)^2 = a^2 + 2ab + b^2\) and \((a - b)^2 = a^2 - 2ab + b^2\). Knowing this helps factor expressions and verify solutions by expanding them back out.

In our exercise, after determining the trinomial is a perfect square \((7x - 2y)^2\), we can expand it to \((7x - 2y)(7x - 2y)\). Each term must be multiplied and simplified to ensure it matches the original expression.

Quadratic expressions have the highest degree of two, typically written as \(ax^2 + bx + c\). They can be solved and factored in different ways, including recognizing patterns or using formulas like the quadratic formula.

A key feature of quadratic expressions is they can often be factored into simpler binomials if the expression is factorable. Perfect square trinomials are special quadratic expressions that can be easily recognized and factored if you spot the pattern.

In our example \(49x^2 - 28xy + 4y^2\), the expression follows the form of a perfect square trinomial, making it straightforward to factor it to \((7x - 2y)^2\). This saves time compared to using other methods, like completing the square or the quadratic formula.