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In algebra, a binomial is a polynomial expression with exactly two terms, often joined by an addition or subtraction operator. For example, \((x+3)\) and \((x+5)\) are both binomials. When you multiply two binomials, as in this exercise, the process involves using each term of the first binomial to multiply every term in the second binomial. This type of multiplication is essential because it lays the foundation for more complex polynomial operations.

To perform binomial multiplication efficiently, ensure that you take each term of the first binomial and multiply it across all terms of the second one. In simpler words, every term has to "meet" every other term, leading to new expressions. The outcome typically results in a trinomimal, but could be more complex depending on the initial binomials involved.

The distributive property is a valuable algebraic principle used to simplify expressions and multiply polynomials. It states that \(a(b+c) = ab + ac\). In the context of our example, it helps to distribute each term of one polynomial to every term of another. This is why the process is sometimes known as "distributing".

When solving the expression \((x+3)(x+5)\), apply distribution by first multiplying the term \(x\) from the first binomial by every term in the second, resulting in \(x^2 + 5x\). Next, distribute \(3\) similarly, giving \(3x + 15\). This method ensures that no term is left out of the multiplication process and sets the stage for the next step in solving the problem.

After multiplying the binomials and using the distributive property, the next step is to combine like terms. This means you should add together terms in the resulting expression that have the same variable raised to the same power.

In our example, after distributing you'll have an expression: \(x^2 + 5x + 3x + 15\). Notice that the terms \(5x\) and \(3x\) both have the variable \(x\). These are considered like terms because they have the same variable and exponent. When you combine them by adding, you get \(8x\). This gives you a simplified expression: \(x^2 + 8x + 15\).

Remember, combining like terms helps simplify expressions, making them easier to understand and work with. It's an invaluable skill in algebra that streamlines solving equations and manipulating polynomials.