Answer :

To factor the expression [tex]\( 12s^2 - 147 \)[/tex], we can start by looking for a common factor in both terms.

1. Identify the Greatest Common Factor (GCF):
The expression [tex]\( 12s^2 - 147 \)[/tex] consists of the terms [tex]\( 12s^2 \)[/tex] and [tex]\(-147\)[/tex]. First, let's find the greatest common factor of the coefficients 12 and 147. The GCF of 12 and 147 is 3.

2. Factor Out the GCF:
Factor out 3 from the expression:
[tex]\[
12s^2 - 147 = 3(4s^2 - 49)
\][/tex]

3. Recognize a Difference of Squares:
The expression inside the parentheses, [tex]\( 4s^2 - 49 \)[/tex], is a difference of squares. A difference of squares can be factored using the pattern [tex]\( a^2 - b^2 = (a - b)(a + b) \)[/tex].

4. Apply the Difference of Squares Formula:
In [tex]\( 4s^2 - 49 \)[/tex], notice that [tex]\( 4s^2 = (2s)^2 \)[/tex] and [tex]\( 49 = 7^2 \)[/tex]. Therefore, we can write:
[tex]\[
4s^2 - 49 = (2s)^2 - 7^2
\][/tex]

Applying the difference of squares formula, we have:
[tex]\[
(2s)^2 - 7^2 = (2s - 7)(2s + 7)
\][/tex]

5. Combine the Steps:
Now, substitute back into the expression with the GCF factored out:
[tex]\[
3(4s^2 - 49) = 3 \times (2s - 7) \times (2s + 7)
\][/tex]

Thus, the factored form of [tex]\( 12s^2 - 147 \)[/tex] is:
[tex]\[
3(2s - 7)(2s + 7)
\][/tex]

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