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To simplify a polynomial expression, combining like terms is essential. Like terms are terms that have the same variable raised to the same power. In our example, the expression starts as \(2a^2 - 11a + 12 - 32 + 2a^2\).

Notice that \(2a^2\) appears twice, so they can be combined. Here's how it works:

• Add the coefficients of the same terms: \(2a^2 + 2a^2 = 4a^2\)

This process minimizes the terms into a simpler format.

Remember: Only terms with the exact same variables and exponents can be combined. If a term has a different variable or exponent, like \(a\) versus \(a^2\), they aren't like terms and remain separate in the expression.

Combining constants also follows the same rule. In the expression, \(12\) and \(-32\) both lack variables and can be combined to get \(-20\). This leads us to the simplified expression \(4a^2 - 11a - 20\).

A quadratic expression is one where the highest exponent of the variable is 2. This identifies its degree as a quadratic polynomial. In \(4a^2 - 11a - 20\), the term \(4a^2\) shows it's quadratic because the variable \(a\) is squared.

Quadratics often take the form \(ax^2 + bx + c\), where \(a, b,\) and \(c\) are constants.

• \(a\) is the coefficient of the quadratic term.
• \(b\) is the coefficient of the linear term.
• \(c\) is the constant term.

In our expression, \(a = 4, b = -11,\) and \(c = -20\).

Quadratic expressions can sometimes be factored, but not all can be. Checking with factoring techniques like the quadratic formula or trial and error helps, though a neatly divisible factor wasn't found in our expression. So, it stays as it is.

Rearranging terms is a helpful strategy to make combining like terms easier. By grouping similar terms together initially, it sets the stage for smoother simplification.

Given expression: \(2a^2 - 11a + 12 - 32 + 2a^2\).

Step-by-Step Rearrangement:

• Identify and gather like terms: \((2a^2 + 2a^2) - 11a + (12 - 32)\)
• Write similar components next to each other for ease in the next steps.

This organized structure makes the process of combining terms efficient and reduces errors.

Remember, the order of adding or subtracting terms doesn't change the value of the expression due to the commutative property of addition. Thus, reordering terms is a useful tool in simplification.