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When working with polynomials, it's important to first write them in standard form. This means arranging the terms by the degrees (or exponents) of the variable. The highest exponent comes first, followed by the next highest, and so on. For example, the polynomial given is initially written as \(30 - 4x - 2x^2\). In standard form, this becomes \(-2x^2 - 4x + 30\). Writing in standard form helps to easily identify patterns and relationships between the terms.

Factoring out the greatest common factor (GCF) is a key step in simplifying polynomials. The GCF of a set of terms is the largest factor that divides each of the terms. For our example, the terms \(-2x^2\), \(-4x\), and \(30\) share a GCF of \(-2\). By factoring out \(-2\), we simplify the polynomial to \(-2(x^2 + 2x - 15)\).
Steps to identify and factor out the GCF:

• Identify the GCF of all terms.
• Divide each term by the GCF.
• Factor the GCF outside of the polynomial.

A quadratic expression is a polynomial where the highest degree of the variable is 2. It's usually in the form \(ax^2 + bx + c\). In our example, the simplified quadratic is \(x^2 + 2x - 15\).
To factor a quadratic expression, look for two numbers that multiply to the constant term (\(-15\)) and add to the coefficient of the middle term (\(2\)). For \(x^2 + 2x - 15\), these numbers are \(5\) and \(-3\), because \(5 \times (-3) = -15\) and \(5 + (-3) = 2\). Thus, we can factor the quadratic as \((x + 5)(x - 3)\).
Combining this with the GCF factored out earlier, we get the final factored form of the polynomial: \(-2(x + 5)(x - 3)\).
This method is useful because it breaks down complex problems into manageable steps.