To factor the polynomial [tex]\(x^4 y + 8 x^3 y - 6 x^2 y^2 - 48 x y^2\)[/tex] completely, follow these steps:
1. Identify the common factors in all terms:
We start by looking for any common factors among the terms of the polynomial. Each term has both an [tex]\(x\)[/tex] and a [tex]\(y\)[/tex] factor. The greatest common factor (GCF) of all the terms is [tex]\(x y\)[/tex].
2. Factor out the greatest common factor (GCF):
[tex]\[
x^4 y + 8 x^3 y - 6 x^2 y^2 - 48 x y^2 = x y (x^3 + 8 x^2 - 6 x y - 48 y)
\][/tex]
3. Look at the remaining polynomial inside the parentheses and factor further:
We now focus on factoring the cubic polynomial [tex]\(x^3 + 8 x^2 - 6 x y - 48 y\)[/tex]. We notice that this polynomial can be rearranged for better clarity or factored by grouping.
4. Rewriting and factoring by grouping:
Group terms to facilitate factoring:
[tex]\[
x^3 + 8 x^2 - 6 x y - 48 y = (x^3 + 8 x^2) + (-6 x y - 48 y)
\][/tex]
Factor each group:
[tex]\[
x^2 (x + 8) - 6 y (x + 8)
\][/tex]
Notice that [tex]\((x + 8)\)[/tex] is a common factor:
[tex]\[
(x^2 - 6 y)(x + 8)
\][/tex]
5. Combine all the factors:
Incorporate the previously factored [tex]\(xy\)[/tex]:
[tex]\[
x y (x^2 - 6 y) (x + 8)
\][/tex]
So, the complete factored form of the given polynomial [tex]\( x^4 y + 8 x^3 y - 6 x^2 y^2 - 48 x y^2 \)[/tex] is:
[tex]\[
x y (x + 8) (x^2 - 6 y)
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{x y (x + 8) (x^2 - 6 y)}
\][/tex]
