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When multiplying binomials, the FOIL method is an easy way to ensure every term is accounted for. FOIL is an acronym that stands for First, Outer, Inner, and Last. Let's break it down:

1. **First:** Multiply the first terms of each binomial: \(10a \times 3a = 30a^2\)
2. **Outer:** Multiply the outer terms of each binomial: \(10a \times -3 = -30a\)
3. **Inner:** Multiply the inner terms of each binomial: \(-1 \times 3a = -3a\)
4. **Last:** Multiply the last terms of each binomial: \(-1 \times -3 = 3\)

By following these steps, we make sure we multiply each part of one binomial with every part of the other binomial. The results of each of these multiplications form the new expression.

The distributive property is a fundamental algebraic property that allows us to multiply a single term by each term in a binomial. The principle is simple and essential for understanding more complex algebraic concepts.

In our example, we distribute each term in the first binomial \((10a - 1)\) over the second binomial \((3a - 3)\). This means:

\((10a \times 3a) + (10a \times -3) + (-1 \times 3a) + (-1 \times -3)\)

Each term from the first binomial is multiplied by each term in the second binomial. First, we get \(10a \times 3a = 30a^2\). Then, we get \(10a \times -3 = -30a\). Next, \(-1 \times 3a = -3a\) and finally, \(-1 \times -3 = 3\). Understanding this distribution process is key to mastering binomial multiplication.

Once the binomials are multiplied using the FOIL method and distributive property, you'll often end up with like terms—terms that have the same variable raised to the same power. Combining these like terms simplifies the expression.

In our problem, the like terms are \(-30a\) and \(-3a\). We combine them by simply adding the coefficients (numbers in front of the variables):

\[-30a - 3a = -33a\]

So, after combining, our expression looks like:
\[30a^2 - 33a + 3\]
This final step is crucial as it ensures the polynomial is written in its simplest form. It’s like tidying up your math work to make it neat and comprehensible.