Answer :

Sure, let's factor the polynomial [tex]\(6x^3 + 48y^3\)[/tex] step-by-step.

### Step 1: Factor out the Greatest Common Factor (GCF)

First, we need to find the greatest common factor of the coefficients [tex]\(6\)[/tex] and [tex]\(48\)[/tex]. The GCF of [tex]\(6\)[/tex] and [tex]\(48\)[/tex] is [tex]\(6\)[/tex].

So we can factor out [tex]\(6\)[/tex] from each term:

[tex]\[
6x^3 + 48y^3 = 6(x^3 + 8y^3)
\][/tex]

### Step 2: Recognize the Sum of Cubes

Next, we observe that the expression inside the parentheses, [tex]\(x^3 + 8y^3\)[/tex], is a sum of cubes. Recall the sum of cubes formula:

[tex]\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\][/tex]

Here, [tex]\(a = x\)[/tex] and [tex]\(b = 2y\)[/tex] because [tex]\(8y^3 = (2y)^3\)[/tex].

### Step 3: Apply the Sum of Cubes Formula

Using [tex]\(a = x\)[/tex] and [tex]\(b = 2y\)[/tex], apply the sum of cubes formula:

[tex]\[
x^3 + (2y)^3 = (x + 2y)\left(x^2 - x(2y) + (2y)^2\right)
\][/tex]

Simplifying inside the parentheses:

[tex]\[
x^2 - 2xy + 4y^2
\][/tex]

### Step 4: Combine All Parts

Putting it all together, we factor the original polynomial completely:

[tex]\[
6(x^3 + 8y^3) = 6\left((x + 2y)(x^2 - 2xy + 4y^2)\right)
\][/tex]

### Final Factored Form

So the fully factored form of the polynomial [tex]\(6x^3 + 48y^3\)[/tex] is:

[tex]\[
6(x + 2y)(x^2 - 2xy + 4y^2)
\][/tex]

This is the detailed step-by-step solution for factoring the given polynomial.

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