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A perfect square trinomial is an algebraic expression that can be factored into a binomial multiplied by itself. The general form of a perfect square trinomial is \(x^2 + 2xy + y^2\), where \(x\) and \(y\) are terms from the binomial factors. The key identifying feature of a perfect square trinomial is that the first and last terms are square numbers, and the middle term is twice the product of the roots of the perfect squares.

To determine if a trinomial is a perfect square, you should first check if both the first and last terms are squares of some expression, and then verify if the middle term is exactly twice the product of the expressions whose squares produce the first and last terms. In our exercise, the first term is \(25 a^{2}\), which is the square of \((5a)\), the last term is \(9 b^{2}\), the square of \((3b)\), and the middle term \(30ab\) fits the pattern, as it is twice the product of \((5a)\) and \((3b)\).

Therefore, the given trinomial \(25 a^{2}+30 a b+9 b^{2}\) is indeed a perfect square and can be expressed as \((5a+3b)^{2}\). This format of factorization simplifies complex expressions and is particularly useful in solving quadratic equations.

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. It represents a particular value or set of values. Trinomials, like in our example, are a type of algebraic expression with three terms. These expressions become the foundation for solving equations and can describe relationships between different quantities.

Understanding algebraic expressions is critical in mathematics because they are used to model real-world situations and to find unknown values. They also provide a way to succinctly communicate mathematical ideas without using a lot of words. For instance, when we look at the expression from the exercise, \(25 a^{2}+30 a b+9 b^{2}\), it tells us that we have a combination of squared terms and a product term, enabling us to visualize this relationship quickly.

The process of factoring algebraic expressions involves breaking down the expressions into simpler, irreducible factors that, when multiplied together, give the original expression. This process is analogous to finding the prime factorization of a number. Factoring is an essential skill because it allows for the simplification of expressions and the solving of equations.

In the context of trinomials, factoring often involves finding two binomials that produce the original trinomial when multiplied. This concept is particularly important for solving quadratic equations, where the trinomial is set equal to zero. In addition, factoring is used in calculus to simplify derivatives and integrals. An example from our exercise is factoring \(25 a^{2}+30 a b+9 b^{2}\) into \((5a+3b)^{2}\), which simplifies the expression and reveals its underlying structure.