`(20u^(3)v6(2)-15u^(2)v)/(10u^(4)v+30u^(3)v^(3))`
Which of the following is the reduced form of the expression above?

Options

A. `(5uv)/(40u^(7)v^(4))`
B. `(2v-1)/(u+2uv^(2))`
C. `(4uv-3)/(2u^(2)+2uv^(2))`
D. `(2uv-3uv^(2))/(u^(2)+6)`

Explanation

To simplify the expression \((20u^{3}v^{2} - 15u^{2}v) / (10u^{4}v + 30u^{3}v^{3})\), we will follow these steps:

### Step 1: Identify the Numerator and Denominator
The expression can be rewritten as:
\[ \frac{20u^{3}v^{2} - 15u^{2}v}{10u^{4}v + 30u^{3}v^{3}} \]

### Step 2: Factor the Numerator
In the numerator \(20u^{3}v^{2} - 15u^{2}v\), we need to find the greatest common factor (GCF):
- The GCF of the coefficients \(20\) and \(15\) is \(5\).
- The GCF of \(u^{3}\) and \(u^{2}\) is \(u^{2}\).
- The GCF of \(v^{2}\) and \(v\) is \(v\).

Thus, the GCF of the numerator is \(5u^{2}v\). We can factor it out:
\[ 20u^{3}v^{2} - 15u^{2}v = 5u^{2}v(4u v - 3) \]

### Step 3: Factor the Denominator
In the denominator \(10u^{4}v + 30u^{3}v^{3}\), we again find the GCF:
- The GCF of \(10\) and \(30\) is \(10\).
- The GCF of \(u^{4}\) and \(u^{3}\) is \(u^{3}\).
- The GCF of \(v\) and \(v^{3}\) is \(v\).

Thus, the GCF of the denominator is \(10u^{3}v\). We can factor it out:
\[ 10u^{4}v + 30u^{3}v^{3} = 10u^{3}v(u + 3v^{2}) \]

### Step 4: Rewrite the Expression
Now we can rewrite the expression with the factored forms:
\[ \frac{5u^{2}v(4u v - 3)}{10u^{3}v(u + 3v^{2})} \]

### Step 5: Cancel Common Factors
We can cancel the common factors in the numerator and denominator:
- \(5\) in the numerator and \(10\) in the denominator simplifies to \(1/2\).
- \(u^{2}\) in the numerator and \(u^{3}\) in the denominator simplifies to \(1/u\).
- \(v\) in the numerator and \(v\) in the denominator cancels out.

After canceling, we have:
\[ \frac{(4u v - 3)}{2u(u + 3v^{2})} \]

### Step 6: Final Expression
Thus, the reduced form of the expression is:
\[ \frac{4u v - 3}{2u^{2} + 6u v^{2}} \]

### Conclusion
The reduced form of the expression is \(\frac{4u v - 3}{2u^{2} + 6u v^{2}}\).