To factor the given expression [tex]\(x^2 y - 2 x y - 24 y\)[/tex], let's go through the steps systematically.
1. Identify the common factor in the terms:
Each term in the expression [tex]\(x^2 y - 2 x y - 24 y\)[/tex] includes a [tex]\(y\)[/tex]:
[tex]\[
x^2 y - 2 x y - 24 y = y(x^2 - 2x - 24)
\][/tex]
2. Factor the quadratic expression:
Now, we need to factor the quadratic inside the parentheses: [tex]\(x^2 - 2x - 24\)[/tex].
3. Find the factors of the quadratic:
We look for two numbers whose product is [tex]\(-24\)[/tex] (the constant term) and whose sum is [tex]\(-2\)[/tex] (the coefficient of the linear term [tex]\(x\)[/tex]).
The numbers that satisfy these conditions are [tex]\( -6 \)[/tex] and [tex]\( 4 \)[/tex], since:
[tex]\[
-6 \times 4 = -24 \quad \text{and} \quad -6 + 4 = -2
\][/tex]
4. Write the quadratic as a product of binomials:
Given the factors, we can write:
[tex]\[
x^2 - 2x - 24 = (x - 6)(x + 4)
\][/tex]
5. Combine the common factor with the factored quadratic:
Substitute back the factor [tex]\(y\)[/tex]:
[tex]\[
y(x^2 - 2x - 24) = y(x - 6)(x + 4)
\][/tex]
Therefore, the completely factored form of the expression [tex]\(x^2 y - 2 x y - 24 y\)[/tex] is:
[tex]\[
y(x - 6)(x + 4)
\][/tex]
So, the correct choice is:
[tex]\[
\boxed{y(x+4)(x-6)}
\][/tex]
