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The distributive property is a fundamental concept in algebra that allows us to simplify expressions and equations efficiently. It states that multiplying a single term by a sum within parentheses is the same as multiplying each addend individually and then adding the results together.

This property is often expressed as:

• \( a(b + c) = ab + ac \)

For our specific problem, we apply the distributive property to simplify the left side of the equation \( 5(x + 2) \). We do this by multiplying the \( 5 \) by each term within the parentheses:

- \( 5 \times x = 5x \)
- \( 5 \times 2 = 10 \)

After distributing, the expression becomes \( 5x + 10 \). This step is crucial for setting up the equation to be solved.

Sometimes in algebra, after simplifying the equation, we find that there are no solutions. These are known as no solution equations. Such situations occur when the simplified equation leads to a contradictory statement. For example, a statement like \( 10 = -2 \) indicates no possible value of \( x \) can satisfy the equation.

Within this problem, after using the distributive property and simplifying the expression, we set both sides equal and eliminated like terms. The resulting equation, \( 10 = -2 \), clearly makes no logical sense because "10" does not equal "-2." This contradiction means the equation has no solution.

In algebra, equations with no solutions might seem strange at first, but they tell us something important: there are instances where no real number satisfies the equation. Recognizing these types of equations is an essential skill in algebra.

Simplifying expressions is a key step in solving algebraic equations. It involves reducing expressions to their simplest form, making them easier to handle and solve.

In our exercise, we simplify each side of the equation as follows:

• First, apply the distributive property on the left side to expand \( 5(x + 2) \), resulting in \( 5x + 10 \).
• On the right side, we had \( 5x - 2 \).

Next, we set both expressions equal to each other: \( 5x + 10 = 5x - 2 \). We then subtract \( 5x \) from both sides to further simplify, eliminating like terms:

\( 5x + 10 - 5x = 5x - 2 - 5x \)

This simplifies to: \( 10 = -2 \).

In this example, simplifying expressions reveals an important outcome: identifying that the equation has no solutions. Simplification is not just about reducing clutter; it helps us uncover the true nature of the algebraic equation we're tackling.