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The distributive property is an essential algebraic rule that allows us to multiply a single term by each term within a parenthesis. For the given problem,
we apply the distributive property by multiplying each term from the first polynomial, \((3x+4)\), with each term from the second polynomial, \((2x+6)\). This results in four individual products: \(3x\) multiplied by \(2x\), \(3x\) multiplied by 6, 4 multiplied by \(2x\), and finally 4 multiplied by 6.
The initial expansion looks like this: \[(3x+4)(2x+6) = 3x(2x) + 3x(6) + 4(2x) + 4(6)\].
Remember, when using the distributive property, ensure you multiply each term correctly to avoid any errors in your final answer.

After using the distributive property, the next step is to simplify the expression further by combining like terms. Like terms are terms that have the same variables raised to the same powers. In our problem, after distributing we are left with: \[6x^2 + 18x + 8x + 24\].
Here we notice that \(18x\) and \(8x\) are like terms because they both have the variable 'x' raised to the same power.
To combine these, we simply add their coefficients: \(18 + 8\) to get 26, and then rewrite the expression as \[6x^2 + 26x + 24\].
It's important not to rush this step and carefully look for terms that can be combined to simplify the expression as much as possible.

Polynomial multiplication involves distributing each term of one polynomial with every term of another polynomial. It is a process you'll find yourself using often in algebra.
In our example, the original expression \((3x+4)(2x+6)\) represents the multiplication of two binomials. The method used to multiply these polynomials is sometimes referred to as FOIL (First, Outer, Inner, Last), relating to the terms' positions in each binomial.
However, it's essential to remember that FOIL is just a shortcut for the distributive property applied multiple times. The final result from our problem demonstrates polynomial multiplication: \[6x^2 + 26x + 24\].
Remember to always multiply coefficients with coefficients and variables with variables, using exponent rules when necessary (for example, \(x\'x\) becomes \(x^2\)).