Answer :

Sure! Let's factor the polynomial [tex]\(48x^3 + 112x^2 - 3x - 7\)[/tex] completely.

Step 1: Group terms to find a common factor.
We start by grouping the terms in pairs:
[tex]\[
(48x^3 + 112x^2) + (-3x - 7)
\][/tex]

Step 2: Factor out the greatest common factor (GCF) from each group.
The GCF of the first group [tex]\(48x^3 + 112x^2\)[/tex] is [tex]\(16x^2\)[/tex], so we factor that out:
[tex]\[
16x^2(3x + 7)
\][/tex]

In the second group [tex]\(-3x - 7\)[/tex], there is no common factor other than 1, so we factor out -1:
[tex]\[
-1(3x + 7)
\][/tex]

Now, our expression is:
[tex]\[
16x^2(3x + 7) - 1(3x + 7)
\][/tex]

Step 3: Factor by grouping.
Notice both terms have a common binomial factor [tex]\((3x + 7)\)[/tex]:
[tex]\[
(3x + 7)(16x^2 - 1)
\][/tex]

Step 4: Factor any further difference of squares.
The expression [tex]\(16x^2 - 1\)[/tex] is a difference of squares:
[tex]\[
16x^2 - 1 = (4x)^2 - 1^2
\][/tex]

We use the difference of squares formula, [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex]:
[tex]\[
16x^2 - 1 = (4x - 1)(4x + 1)
\][/tex]

Step 5: Write the factored form.
Now that we have [tex]\(16x^2 - 1\)[/tex] factored completely, the full expression becomes:
[tex]\[
(3x + 7)(4x - 1)(4x + 1)
\][/tex]

Therefore, the completely factored form of the polynomial [tex]\(48x^3 + 112x^2 - 3x - 7\)[/tex] is [tex]\((3x + 7)(4x - 1)(4x + 1)\)[/tex].

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