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Understanding the distribution of terms is essential when multiplying polynomials. When faced with an expression like \(5 x^{2}-4\right)\times\left(3 x^{2}-7\)), we multiply each term in the first polynomial by each term in the second polynomial. This ensures that we account for every combination that arises from the product.

For example, we begin by taking \(5x^2\) and multiplying it by \(3x^2\) and \( -7\), which gives us \(15x^4\) and \( -35x^2\), respectively. The reason behind this is that we are following the distributive property which states that \(a*(b+c) = a*b + a*c\). This initial step sets up the framework for our polynomial multiplication.

In the next phase, we take the second term of the first polynomial, which is \( -4\), and distribute it across the second polynomial. This step by step method ensures we cover all the necessary terms in the multiplication process without omitting any.

After distributing terms, it's important to simplify the expression by combining like terms. Like terms are terms that have the same variable raised to the same power. In our exercise, the terms \( -35x^2\) and \( -12x^2\) are like terms because both are coefficients of \(x^2\).

To combine them, simply add or subtract the coefficients as appropriate. In this case, we subtract \(35\) and \(12\), resulting in \( -47x^2\), which streamlines our polynomial to \(15x^4 - 47x^2 + 28\). Remember, only coefficients can be combined; the variable part remains unchanged. Through this process, we achieve a more compact and simplified final expression.

At its core, a polynomial expression is a mathematical expression consisting of variables, coefficients, and exponents, arranged in a way that the exponents are non-negative integers. In the context of our exercise, \(15x^4 - 47x^2 + 28\) is a polynomial expression.

The structure of a polynomial holds significant information. For instance, the highest power of the variable (in our case, \( x^4\)) tells us the degree of the polynomial, which is 4 in this example. A polynomial's degree is a crucial characteristic that affects many aspects of its behavior, such as the number of roots it can have.

Using standard form, where terms are ordered from the highest degree to the lowest, allows for a more intuitive understanding and facilitates operations such as adding, subtracting, or multiplying polynomials. Recognizing the anatomy of polynomial expressions is a valuable skill in algebra that underpins a myriad of more advanced mathematical concepts and procedures.