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Simplifying a fraction to its lowest terms means reducing it to its simplest form, where the numerator (the top number) and denominator (the bottom number) are as small as possible and have no common factors other than 1. To achieve this, we divide both the numerator and the denominator by their greatest common divisor (GCD). In the sample problem, after multiplying the fractions, the resulting fraction is \(\frac{2x^2}{15}\). Here, since \(x^2\) and 15 do not share any common factors other than 1, there is no need to simplify further, and the fraction is already in its lowest terms.

However, what if the product had common factors? For instance, if we had \(\frac{4x^2}{16}\), we could divide both numerator and denominator by 4, the GCD, resulting in \(\frac{x^2}{4}\). Always check for common factors to ensure fractions are simplified to lowest terms for the most accurate and refined expression in mathematics.

In any multiplication of fractions, the numerators of the given fractions need to be multiplied together to find the numerator of the product. This means that we take the top numbers of the fractions and multiply them as if they were integers. Let's consider our example where we have \(x^{2}\) and 2 as our numerators. We perform the multiplication \(x^2 \times 2\) which results in \(2x^2\). This will be the new numerator of our fraction after multiplication. It's important to treat any variables or exponents as part of the multiplication process, applying the laws of exponents and arithmetic accordingly to ensure the result is simplified correctly.

Similarly, multiplying the denominators of fractions requires us to take the bottom numbers, the denominators, and multiply them together. In our example, the denominators were 5 and 3. We multiply these to get \(5 \times 3 = 15\). This product, 15, becomes the denominator of our new fraction. It is critical to remember that unlike adding or subtracting fractions, where a common denominator is necessary, multiplying fractions simply requires direct multiplication of the denominators, without any need for finding a common value. However, after multiplying, always check to see if the new denominator can be simplified in conjunction with the numerator, as part of converting the fraction into its lowest terms.