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To solve quadratic equations, one often needs to expand products of binomials. Expanding an equation might look intimidating at first, but it's a systematic process. It involves using the distributive property, also known as the FOIL method for binomials. Here, each term in the first expression is multiplied by each term in the second expression.

For example, with the equation
\[ x(x + 5) = 2(x + 5) \],
expanding the left-hand side, we multiply 'x' by 'x' and by '5' to get
\[ x^2 + 5x \].
Similarly, expanding the right-hand side, we multiply '2' by 'x' and by '5' to get
\[ 2x + 10 \].
This process is crucial because it transforms the equation into a form where like terms are visible and can be combined, leading us to a standard quadratic equation.

Rearranging equations is essential for solving quadratic equations. It's about moving terms from one side of the equation to the other to form a standard quadratic equation, which is in the form
\[ ax^2 + bx + c = 0 \].
The goal of rearranging is to isolate the variable terms on one side and the constants on the other.

Take the expanded equation
\[ x^2 + 5x = 2x + 10 \].
Here, our goal is to move the '2x' and '10' to the left-hand side, which results in
\[ x^2 + 5x - 2x - 10 = 0 \].
Now, the equation is in a form where we can easily spot like terms and combine them to solve for 'x'. Being comfortable with rearranging terms is a pivotal step toward mastering quadratic equations.

Once an equation is expanded and rearranged, the next step in solving a quadratic equation is combining like terms. Like terms are terms that have the same variable raised to the same power. By combining them, we simplify the equation further.

For instance, in the rearranged equation
\[ x^2 + 5x - 2x - 10 = 0 \],
the terms '5x' and '-2x' are like terms as they both contain the variable 'x' to the first power. Combine them to get '3x'.

By doing so, we now have the simplified quadratic equation
\[ x^2 + 3x - 10 = 0 \].
This equation is now easier to solve, either by factoring, if possible, or using quadratic formula. Understanding how to combine like terms accurately ensures a smoother journey to finding the roots of quadratic equations.