Answer :
We start with the expression
[tex]$$27x^2 - 12.$$[/tex]
Step 1. Factor out the greatest common factor (GCF). Both terms are divisible by [tex]$3$[/tex], so we have
[tex]$$27x^2 - 12 = 3(9x^2 - 4).$$[/tex]
Step 2. Notice that the expression inside the parentheses, [tex]$9x^2 - 4$[/tex], is a difference of two squares. In fact,
[tex]$$9x^2 = (3x)^2 \quad \text{and} \quad 4 = 2^2.$$[/tex]
Thus, we can write
[tex]$$9x^2 - 4 = (3x)^2 - 2^2.$$[/tex]
Step 3. Use the difference of squares formula, which states that
[tex]$$a^2 - b^2 = (a - b)(a + b),$$[/tex]
by letting [tex]$a = 3x$[/tex] and [tex]$b = 2$[/tex], we obtain
[tex]$$9x^2 - 4 = (3x - 2)(3x + 2).$$[/tex]
Step 4. Substitute this back into the expression from Step 1:
[tex]$$27x^2 - 12 = 3(3x - 2)(3x + 2).$$[/tex]
Thus, the fully factored form of the expression is
[tex]$$\boxed{3(3x - 2)(3x + 2)}.$$[/tex]
[tex]$$27x^2 - 12.$$[/tex]
Step 1. Factor out the greatest common factor (GCF). Both terms are divisible by [tex]$3$[/tex], so we have
[tex]$$27x^2 - 12 = 3(9x^2 - 4).$$[/tex]
Step 2. Notice that the expression inside the parentheses, [tex]$9x^2 - 4$[/tex], is a difference of two squares. In fact,
[tex]$$9x^2 = (3x)^2 \quad \text{and} \quad 4 = 2^2.$$[/tex]
Thus, we can write
[tex]$$9x^2 - 4 = (3x)^2 - 2^2.$$[/tex]
Step 3. Use the difference of squares formula, which states that
[tex]$$a^2 - b^2 = (a - b)(a + b),$$[/tex]
by letting [tex]$a = 3x$[/tex] and [tex]$b = 2$[/tex], we obtain
[tex]$$9x^2 - 4 = (3x - 2)(3x + 2).$$[/tex]
Step 4. Substitute this back into the expression from Step 1:
[tex]$$27x^2 - 12 = 3(3x - 2)(3x + 2).$$[/tex]
Thus, the fully factored form of the expression is
[tex]$$\boxed{3(3x - 2)(3x + 2)}.$$[/tex]
