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In mathematics, function substitution involves replacing the variable in a function with another expression. This helps evaluate the function at different points.

For instance, with the function given as \(g(x) = 4x - 2\), substituting \(x + h\) means we replace every appearance of \(x\) with \(x + h\).

Therefore, \(g(x + h)\) becomes:

\[g(x+h) = 4(x+h) - 2\]

By performing this step, you transform the function to reflect a new input, which is crucial for further calculations.

Simplification is the process of making a mathematical expression more manageable or easier to understand by combining like terms and performing basic arithmetic operations.

During simplification, especially in polynomial expressions, combine terms that have the same variable raised to the same power. For the expression we obtained in the previous step, \(g(x + h) = 4(x + h) - 2\), we simplify:

\[g(x+h) = 4x + 4h - 2\]

By distributing the 4 across the terms inside the parenthesis and then combining like terms where possible, the expression becomes more straightforward and useful for subsequent steps.

Polynomials are expressions that involve variables raised to whole number powers and their coefficients. In the functions given, \(f(x) = 3x^2 - x\) and \(g(x) = 4x - 2\), both are polynomials.

Polynomials can be as simple as \(g(x)\), a linear polynomial with a degree of 1, or more complex like \(f(x)\), a quadratic polynomial with a degree of 2.

When working with polynomials, understanding their structure helps in both substitution and simplification. For instance, when subtracting one polynomial from another, as in the final step of our exercise:

\[g(x+h) - g(x) = (4x + 4h - 2) - (4x - 2)\], we can immediately see the like terms \(4x\) and \(-2\) that will cancel each other out:

\[g(x+h) - g(x) = 4h\]

This ability to recognize and manipulate polynomials is essential in algebra and calculus.