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The distributive property is a fundamental concept in algebra that allows us to simplify expressions where one term is multiplied by a sum or difference inside parentheses. In the given exercise, we use the distributive property to expand the expression \( (s+8)(s+9) \).

First, we distribute the term \( s \) to both terms in the second binomial: \( s \times s + s \times 9 \). This gives us \[ s^2 + 9s \].

Next, we distribute the term \( 8 \) to both terms in the second binomial: \[ 8 \times s + 8 \times 9 \]. This results in \[ 8s + 72 \].

By applying the distributive property correctly, we can break down complex multiplications into simpler, more manageable steps. This is extremely useful in solving polynomial expressions and equations.

Combining like terms is the process of merging terms in an expression that have the same variable raised to the same power. For our simplified expressions, after distributing, we obtained two sets of terms: \[ s^2 + 9s \] and \[ 8s + 72 \].

To combine the like terms, we look for terms that have the same variable component. In our case, those are \( 9s \) and \( 8s \).

Adding these together, we get \[ 9s + 8s = 17s \].

After combining, our expression now looks like this: \[ s^2 + 17s + 72 \]. This step is crucial because it simplifies the expression further, making it easier to interpret and solve.

Polynomial simplification involves reducing a polynomial to its simplest form by combining like terms and ensuring that there are no unnecessary parentheses. In the given example, we started with the expression \( (s+8)(s+9) \).

By applying the distributive property and combining like terms, we eventually simplified it to \[ s^2 + 17s + 72 \].

Here are the steps we followed:

• Distributed \( s \) to \( s+9 \), resulting in \[ s^2 + 9s \].
• Distributed \( 8 \) to \( s+9 \), resulting in \[ 8s + 72 \].
• Combined \( 9s \) and \( 8s \) to get \[ 17s \].

The final simplified polynomial is now clean and free of any unnecessary complexity. This simplified form is vital for solving equations, graphing functions, and further algebraic manipulations.