### Part A: Finding a Common Factor
The given expression is [tex]\(30x^4y^3 - 12x^3y\)[/tex].
A common factor for this expression, containing at least one variable and having a coefficient other than 1, is [tex]\(6x^3y\)[/tex].
### Part B: Explanation of Finding the Common Factor
To determine the common factor:
1. Coefficients: First, we find the greatest common divisor (GCD) of the coefficients 30 and 12. The GCD of 30 and 12 is 6.
2. Variable [tex]\(x\)[/tex]: Next, we look at the powers of the variable [tex]\(x\)[/tex] in each term. The first term has [tex]\(x^4\)[/tex], and the second term has [tex]\(x^3\)[/tex]. The lowest power of [tex]\(x\)[/tex] common to both terms is [tex]\(x^3\)[/tex].
3. Variable [tex]\(y\)[/tex]: Finally, we consider the powers of the variable [tex]\(y\)[/tex] in each term. The first term has [tex]\(y^3\)[/tex], and the second term has [tex]\(y\)[/tex]. The lowest power of [tex]\(y\)[/tex] common to both terms is [tex]\(y\)[/tex].
Combining these results, we get the common factor [tex]\(6x^3y\)[/tex].
### Part C: Rewriting the Expression Using the Common Factor
To factor the expression [tex]\(30x^4y^3 - 12x^3y\)[/tex] using the common factor [tex]\(6x^3y\)[/tex]:
1. Extract the common factor:
[tex]\[
30x^4y^3 - 12x^3y = 6x^3y \cdot \left( \frac{30x^4y^3}{6x^3y} \right) - 6x^3y \cdot \left( \frac{12x^3y}{6x^3y} \right)
\][/tex]
2. Simplify each term inside the parentheses:
[tex]\[
= 6x^3y \cdot \left( 5xy^2 \right) - 6x^3y \cdot \left( 2 \right)
\][/tex]
3. Combine the simplified terms:
[tex]\[
= 6x^3y \cdot (5xy^2 - 2)
\][/tex]
Thus, the expression [tex]\(30x^4y^3 - 12x^3y\)[/tex] can be factored as:
[tex]\[
30x^4y^3 - 12x^3y = 6x^3y(5xy^2 - 2)
\][/tex]
In conclusion, the common factor of the given expression is [tex]\(6x^3y\)[/tex], and the factored form of the expression is [tex]\(6x^3y(5xy^2 - 2)\)[/tex].
