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The distributive property is a fundamental concept in algebra that allows us to multiply a single term by two or more terms inside a parenthesis. It states that multiplying a number by a sum is the same as multiplying each addend individually and then adding the products together.
When we have an expression like \(a(b+c)\), the distributive property tells us to multiply \(a\) with both \(b\) and \(c\), resulting in \(ab + ac\).
In the given problem, the distributive property is applied in two stages. First, while expanding the binomials \( (p+4)(p-6) \), and then again when we distribute \(p\) across the simplified terms.

A binomial is a polynomial with exactly two terms. Each term in a binomial is separated by either a plus (+) or minus (-) sign. Examples include \(x+3\), \(2a-5\), and \(p+4\). In polynomial multiplication, particularly when multiplying binomials, we often use a method called FOIL (First, Outer, Inner, Last) to apply the distributive property step-by-step.
In the given problem, we start by expanding the binomials \( (p+4)(p-6) \). Here's how we do it using the FOIL method:

• First: Multiply the first terms in each binomial: \(p \times p = p^2\)
• Outer: Multiply the outer terms: \(p \times -6 = -6p\)
• Inner: Multiply the inner terms: \(4 \times p = 4p\)
• Last: Multiply the last terms: \(4 \times -6 = -24\)

Adding these together we get \( p^2 - 6p + 4p - 24\). This can then be simplified.

Simplifying expressions means reducing them to their simplest form. This typically involves combining like terms and performing basic arithmetic operations.
First, we simplify the expression obtained from multiplying the binomials \( (p+4)(p-6) \): \( p^2 - 6p + 4p - 24 \). Notice that \(-6p\) and \(4p\) are like terms because they both have the variable \(p\). We combine them to get \(-2p\). Thus, the expression simplifies to \( p^2 - 2p - 24 \).
Next, we apply the distributive property again by distributing \(p\) over the simplified expression. We multiply \(p \times ( p^2 - 2p - 24 )\), which gives \( p^3 - 2p^2 - 24p \). The final outcome is a simplified polynomial expression.