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Expanding brackets is a foundational skill in algebra that allows us to simplify expressions and equations. To expand brackets means to multiply each term inside the bracket by the factor outside of it.

For example, in the equation from the exercise, we have to deal with two sets of brackets: on the left side of the equation, the bracket \(8(x^2 - 6)\), and on the right side, \(4(x^2 + 13)\). Expanding \(8(x^2 - 6)\) gives us \(8x^2 - 48\), and likewise, expanding \(4(x^2 + 13)\) gives us \(4x^2 + 52\).

It's essential to distribute the multiplication across the terms inside the brackets accurately to avoid any errors in the solution process. This step makes it possible to bring the equation into a form where like terms can be collected, leading us to the next concept.

Once brackets are expanded, the next step is to 'collect like terms'. This means to combine terms that have the same variables raised to the same power. It's a method of simplifying an equation to make it easier to solve.

In our problem, after expanding, we find ourselves with \(8x^2\) and \(4x^2\) as like terms. By subtracting \(4x^2\) from both sides of the equation, we're effectively gathering all the \(x^2\)-terms on one side, which results in \(4x^2\) for this particular problem.

It's also necessary to move constants to the other side, as we do with the -48 and +52, adding them together to get 100. Always ensure that you perform the same operation on both sides of the equation to maintain the balance, which is a fundamental principle in algebra.

Quadratic equations are polynomial equations of degree two. The general form is \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. To find solutions to a quadratic equation, one must isolate \(x\) and determine its values that satisfy the equation.

In our problem, after simplifying, we obtained \(4x^2 = 100\). To solve for \(x\), we divide both sides by 4, obtaining \(x^2 = 25\). The solutions for \(x\) are found by taking the square root of both sides, which gives us \(x = \pm5\).

Remember, the square root operation results in two answers because both \(5^2\) and \( (-5)^2 \) equal 25. Thus, the equation \(x^2 = 25\) has two solutions: \(x = 5\) and \(x = -5\). This illustrates the fundamental concept that a quadratic equation typically has two solutions, which can be equal (in the case of perfect squares) or distinct real or complex numbers.