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Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. They represent quantities without fixed values, often unknown or variable quantities, allowing us to describe general relationships or to perform computations for a variety of inputs.

For example, the expression \(x^{2}+4\) from the exercise \( (x^{2}+4)(x^{2}-4) \) consists of the variable \(x\) raised to the second power, which is combined with the constant number 4 through addition. Such algebraic expressions can be manipulated using various algebraic laws and properties to simplify or expand them, or to prepare them for solving. In this case, our goal is to perform a multiplication to simplify the given pair of expressions.

Factoring polynomials is a critical skill in algebra that involves breaking down a polynomial into simpler 'factor' expressions that multiply together to give the original polynomial.

In the solution of \( (x^{2}+4)(x^{2}-4) \), the key is to recognize that this expression is a special form called the difference of squares. It's a special case which allows factoring based on the structure \(a^{2} - b^{2} = (a + b)(a - b)\), where \(a\) and \(b\) are any algebraic expressions. Here, \(a = x\) and \(b = 2\). By applying the formula of difference of squares, the expression simplifies significantly, allowing us to reach the final expanded polynomial, \(x^{4} - 16\), without the need for complex multiplication steps. Essentially, this form of factoring transforms a polynomial into a product of simpler binomial expressions, making the problem easier to solve.

Exponentiation is an arithmetic operation involving a base and an exponent. It signifies repeated multiplication of the base by itself. \(x^{2}\) indicates that \(x\) is multiplied by itself once.

When multiplying two expressions with the same base, such as \(x^{2}\) and \(x^{2}\), we add the exponents as per the laws of exponents. This operation is key in the given exercise, as the product \(x^{2} \times x^{2}\) results in \(x^{4}\) - this illustrates the rule that \(x^{a} \times x^{b} = x^{a+b}\). Furthermore, when we apply the difference of squares in our problem, we see the exponent on the constant 4; \(4^{2}\) is 16, which is a straightforward calculation. Understanding exponentiation is essential for manipulating expressions and solving algebraic equations involving powers.