Answer :

To factorize the given expression [tex]\(24 x^2 + 6 x y\)[/tex] completely, follow these steps:

1. Identify the Greatest Common Factor (GCF):
Start by identifying the greatest common factor of all terms in the polynomial.

The terms are [tex]\(24 x^2\)[/tex] and [tex]\(6 x y\)[/tex]. The numerical coefficients are 24 and 6, and the highest common factor of these numbers is 6. Both terms also share a common variable, [tex]\(x\)[/tex].

So, the GCF of [tex]\(24 x^2\)[/tex] and [tex]\(6 x y\)[/tex] is [tex]\(6x\)[/tex].

2. Factor out the GCF:
Now, divide each term by the GCF [tex]\(6x\)[/tex]:

[tex]\[ = 24 x^2 \div 6x + 6 x y \div 6x \][/tex]

Simplifying each term:

[tex]\[ = 4x + y \][/tex]

Therefore, when the GCF [tex]\(6x\)[/tex] is factored out from the original expression, it becomes:

[tex]\[ 24 x^2 + 6 x y = 6x(4x + y) \][/tex]

So, the completely factorized form of the expression [tex]\(24 x^2 + 6 x y\)[/tex] is:

[tex]\[ \boxed{6x(4x + y)} \][/tex]