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Coefficients are the numerical parts of terms in a polynomial. When multiplying polynomials, the first step is to multiply these coefficients. It involves taking the coefficients in front of each variable and performing basic multiplication. For instance, in the expression \(9x^2(-4x)\), the coefficients are 9 and -4. Multiplying these gives us \(9 \times (-4) = -36\). The key takeaway when multiplying coefficients is to treat them like regular numbers. Make sure you consider their signs (positive or negative) as this affects the final product. This simple multiplication sets the stage for what's to come.

The variables in a polynomial come with exponents, which indicate how many times the variable is multiplied by itself. When multiplying terms with the same base (in this case, the base is \(x\)), the exponents are added together. The rule for multiplying like bases is: keep the base the same and add the exponents together. For example, in our original expression \(9x^2(-4x)\), we have \(x^2\) and \(x^1\). To multiply these, we add the exponents: \(2 + 1 = 3\). Thus, the resulting term from the multiplication of the variables is \(x^3\). This process simplifies the multiplication of polynomials and follows the laws of exponents. Always check that the variables are the same before adding exponents; otherwise, they cannot be directly combined.

After multiplying the coefficients and adding the exponents, the final step is to combine these results into a single term. This typically involves writing down the resulting coefficient followed by the variable term. From our polynomial multiplication example, the final result becomes \(-36x^3\). Since there are no more like terms to combine with \(-36x^3\), this is the simplified expression.
Combining like terms becomes more relevant in expressions with multiple different terms. In such cases, terms that have the same variable raised to the same power can be combined by adding or subtracting their coefficients. This step ensures that the expression is as simple and understandable as possible. In our example, however, there's only one term left after multiplication, so it stands alone as \(-36x^3\). This process of combining ensures that your final expression runs efficiently and clearly.