Answer :

To factor the expression [tex]\(48x^3 + 112x^2 - 3x - 7\)[/tex] completely, we can follow these steps:

### Step 1: Group the terms
First, we group the terms into pairs:

[tex]\[
(48x^3 + 112x^2) + (-3x - 7)
\][/tex]

### Step 2: Factor out the greatest common factor (GCF) from each group
- First group [tex]\((48x^3 + 112x^2)\)[/tex]:
- The GCF of [tex]\(48x^3\)[/tex] and [tex]\(112x^2\)[/tex] is [tex]\(16x^2\)[/tex].
- Factor out [tex]\(16x^2\)[/tex]:

[tex]\[
48x^3 + 112x^2 = 16x^2(3x + 7)
\][/tex]

- Second group [tex]\((-3x - 7)\)[/tex]:
- The GCF of [tex]\(-3x\)[/tex] and [tex]\(-7\)[/tex] is [tex]\(-1\)[/tex].
- Factor out [tex]\(-1\)[/tex]:

[tex]\[
-3x - 7 = -1(3x + 7)
\][/tex]

### Step 3: Combine the factored groups
Now, notice that both groups contain a common factor of [tex]\((3x + 7)\)[/tex]:

[tex]\[
16x^2(3x + 7) - 1(3x + 7)
\][/tex]

Since [tex]\((3x + 7)\)[/tex] is common to both terms, factor it out:

[tex]\[
(3x + 7)(16x^2 - 1)
\][/tex]

### Step 4: Factor the remaining expression
The expression [tex]\((16x^2 - 1)\)[/tex] is a difference of squares. It can be factored further:

[tex]\[
16x^2 - 1 = (4x)^2 - 1^2 = (4x - 1)(4x + 1)
\][/tex]

### Step 5: Write the completely factored form
Now, combine all the factors:

[tex]\[
48x^3 + 112x^2 - 3x - 7 = (3x + 7)(4x - 1)(4x + 1)
\][/tex]

Therefore, the expression is factored completely into [tex]\((3x + 7)(4x - 1)(4x + 1)\)[/tex].

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