Select an appropriate factoring strategy

Factoring strategy selection involves interpreting an algebraic expression as a product and examining its structure to determine whether to first extract a greatest common factor, use a special-product pattern such as a difference of squares, factor a quadratic trinomial, or apply grouping for suitable four-term expressions. The reasoning includes checking that the proposed factors reproduce the original expression through multiplication; multivariable and higher-degree methods beyond these standard quadratic and polynomial forms are not included.

Detailed Explanation: Select an appropriate factoring strategy

To choose a factoring strategy, first look at the structure of the expression.

Check whether every term has a greatest common factor (GCF).
After taking out the GCF, look for a familiar pattern:
- Difference of squares: $a^2-b^2=(a-b)(a+b)$
- Quadratic trinomial: $ax^2+bx+c$
- Grouping: usually for four terms

Example: Factor $6x^2-54$

Step 1: Find the GCF.

The terms are $6x^2$ and $-54$ . Their greatest common factor is $6$ .

Factor out $6$ :

6x^2-54=6(x^2-9)

Step 2: Examine what remains.

The expression inside the parentheses, $x^2-9$ , has two terms and is a difference of squares:

x^2-9=x^2-3^2

Use the pattern $a^2-b^2=(a-b)(a+b)$ :

x^2-9=(x-3)(x+3)

Therefore,

\boxed{6x^2-54=6(x-3)(x+3)}

Step 3: Check by multiplying.

First multiply the conjugate factors:

(x-3)(x+3)=x^2-9

Then multiply by $6$ :

6(x^2-9)=6x^2-54

This matches the original expression, so the factoring is correct. The key strategy was to take out the GCF first, then recognize the difference-of-squares pattern.

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Factor the Quadratic Equation with Coefficient - Standard Form (With Common Factor) To Answer

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