Sure, let's factorize the given expression [tex]\(20 x^4 y^3 - 30 x^3 y^4\)[/tex].
1. Identify Common Factors:
- First, observe that both terms [tex]\(20 x^4 y^3\)[/tex] and [tex]\(30 x^3 y^4\)[/tex] have common factors. These common factors include numerical and variable parts.
- The numerical coefficients [tex]\(20\)[/tex] and [tex]\(30\)[/tex] have a greatest common divisor (GCD) of [tex]\(10\)[/tex].
- For the variables:
- In terms of [tex]\(x\)[/tex], the smallest power of [tex]\(x\)[/tex] common to both terms is [tex]\(x^3\)[/tex].
- In terms of [tex]\(y\)[/tex], the smallest power of [tex]\(y\)[/tex] common to both terms is [tex]\(y^3\)[/tex].
2. Factor Out the GCD:
- The common factor then is [tex]\(10 x^3 y^3\)[/tex].
Let's factor [tex]\(10 x^3 y^3\)[/tex] out of both terms:
- [tex]\(20 x^4 y^3 = 10 x^3 y^3 \cdot 2x\)[/tex]
- [tex]\(30 x^3 y^4 = 10 x^3 y^3 \cdot 3y\)[/tex]
Hence, we can write the expression as:
[tex]\[ 20 x^4 y^3 - 30 x^3 y^4 = 10 x^3 y^3 (2x) - 10 x^3 y^3 (3y) \][/tex]
3. Combine the Factored Form:
- Combine the factored terms into a single expression:
[tex]\[ 20 x^4 y^3 - 30 x^3 y^4 = 10 x^3 y^3 (2x - 3y) \][/tex]
Therefore, the factorized form of the expression [tex]\(20 x^4 y^3 - 30 x^3 y^4\)[/tex] is:
[tex]\[ 10 x^3 y^3 (2x - 3y) \][/tex]
Looking at the given options, the correct match is:
[tex]\[ \boxed{10 x^3 y^3 (2 x - 3 y)} \][/tex]
