Skill: Select an appropriate factoring strategy

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

1. Check whether every term has a greatest common factor (GCF).
2. 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.