Answer :

To factor out the greatest common factor from the expression [tex]\(12x^3y^3 + 20x^2y^2\)[/tex], let's follow these steps:

1. Identify the coefficients: We have the coefficients 12 and 20.

- The greatest common factor (GCF) of 12 and 20 is 4.

2. Identify the variable parts:

- For [tex]\(x\)[/tex], the terms are [tex]\(x^3\)[/tex] and [tex]\(x^2\)[/tex]. The lowest power here is [tex]\(x^2\)[/tex].

- For [tex]\(y\)[/tex], the terms are [tex]\(y^3\)[/tex] and [tex]\(y^2\)[/tex]. The lowest power here is [tex]\(y^2\)[/tex].

3. Combine the GCFs:

- Combine the GCF of the coefficients with the lowest powers of each variable to get the complete GCF:

[tex]\[
\text{GCF} = 4x^2y^2
\][/tex]

4. Factor out the GCF from the expression:

- Divide each term by the GCF:

[tex]\[
\frac{12x^3y^3}{4x^2y^2} = 3xy
\][/tex]

[tex]\[
\frac{20x^2y^2}{4x^2y^2} = 5
\][/tex]

5. Write the expression as a product of the GCF and the remaining terms:

[tex]\[
12x^3y^3 + 20x^2y^2 = 4x^2y^2(3xy + 5)
\][/tex]

So, the expression factored by the greatest common factor is [tex]\(4x^2y^2(3xy + 5)\)[/tex].

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