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When faced with a quadratic equation like \(x^2 + 6x + 5 = 0\), a key method to find its roots is the quadratic formula. This formula is a guaranteed solution for all quadratic equations, which have the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are known values, and \(a \eq 0\).

The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(\pm\) denotes the two possible values for \(x\). To apply it, one simply needs to identify a, b, and c from the quadratic equation and substitute these values into the formula. In the context of our problem, \(a = 1\), \(b = 6\), and \(c = 5\).

Applying the quadratic formula gives us two potential solutions for \(x\), which addresses our goal of solving the quadratic equation. It's an efficient technique, especially when other methods, such as factoring, are not easily applicable.

Expanding binomials is a useful skill when solving quadratic equations. A binomial is a polynomial with two terms, such as \(x + 3\). When a binomial is squared, we use the formula \( (a + b)^2 = a^2 + 2ab + b^2 \), which is derived from the distributive property, also known as the FOIL method.

In the exercise, we were given \( (x + 3)^2 \), which expands to \( x^2 + 6x + 9 \). It's crucial to correctly expand this to transform the equation into a standard quadratic form before applying other methods such as simplification or the quadratic formula. By mastering binomial expansion, students can greatly simplify the process of solving more complex quadratic equations.

The process of simplifying algebraic expressions is essential in solving equations effectively. It involves combining like terms and eliminating any unnecessary parts of an equation. Like terms are those that have the same variables raised to the same power.

In our problem, after expanding the binomial, we were left with \( x^2 + 6x + 9 - 4 \). Simplifying this involved combining the constant terms, which are \(9\) and \( -4\), to get the simpler form \( x^2 + 6x + 5 = 0\). This step is foundational because it reveals the quadratic equation's true structure, which we can then solve using methods like the quadratic formula. Without simplifying first, it's easy to get lost or make errors as we proceed to solve the equation. Plus, simplification often makes it easier to apply other solution techniques, like factoring or graphing, to find the zeroes of the quadratic function.