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The distributive property is a basic concept in algebra that helps us simplify expressions. It states that multiplying a sum by a number gives the same result as multiplying each addend individually and then adding the products.
This principle is written as: \[ a(b + c) = ab + ac \]

In this exercise, we distribute the constants outside the parentheses (which are 5 and 4) to each term inside. Let's break it down step-by-step:

First, distribute the 5 to every term in the parenthesis:\[ 5(x^2 - 6x + 4) = 5x^2 - 30x + 20 \]

Next, distribute the 4 similarly:\[ 4(x^2 + 3x - 1) = 4x^2 + 12x - 4 \]

By applying the distributive property, we've transformed a more complex expression into simpler pieces that are easier to manage.

To simplify an expression further, we need to combine like terms. Like terms are terms whose variables (and their exponents) are the same. So, in the expression \[5x^2 - 30x + 20 + 4x^2 + 12x - 4 \],
we group terms with \[x^2\], terms with \[x\], and constant terms together:

• Combine the \[x^2\] terms: \[5x^2 + 4x^2 = 9x^2\]
  
  
• Combine the \[x\] terms: \[-30x + 12x = -18x\]
• Combine the constant terms: \[20 - 4 = 16\]

Each grouping helps us move toward a single, simpler expression. Combining like terms reduces multiple terms into fewer terms, making the expression easier to handle, which is very important when solving more complex algebraic equations.

Polynomials are expressions made up of terms that are added or subtracted. Each term consists of a coefficient (a number) and a variable raised to a non-negative integer exponent. For example, \[5x^2 - 30x + 20\] is a polynomial with three terms.

There are several important characteristics of polynomials:

• The degree of a polynomial is the highest exponent of the variable. For instance, in \[9x^2 - 18x + 16\], the degree is 2.
  
• Terms are usually written in descending order of their exponents. This helps to keep polynomials uniformly written and easily comparable.
• Combining like terms and using the distributive property are essential skills for working with polynomials.
  
  In the final step of this exercise, we combine all like terms to get a polynomial in its simplest form: \[9x^2 - 18x + 16\]. This simplified polynomial is much easier to use in further calculations or to better understand the behavior of the function it represents.