If the expression of 2(a^2)(b^2)-7ab+3 is equivalent to (2ab-m)(ab-n), where m and n are constants, what is one possible value of m+n?

Given the expression \( 2a^2b^2 - 7ab + 3 \) and the fact that it can be factored into \( (2ab - m)(ab - n) \), we need to find the values of \( m \) and \( n \) such that the factorization is correct.

### Step 1: Expand the right-hand side
The expression \( (2ab - m)(ab - n) \) expands as follows:

\[
(2ab - m)(ab - n) = 2ab \cdot ab - 2ab \cdot n - m \cdot ab + m \cdot n
\]

This simplifies to:

\[
2a^2b^2 - 2abn - mab + mn
\]

### Step 2: Equate the coefficients
We now compare this expanded expression with the original expression \( 2a^2b^2 - 7ab + 3 \):

\[
2a^2b^2 - 2abn - mab + mn = 2a^2b^2 - 7ab + 3
\]

By equating the coefficients of like terms:

1. The coefficient of \( ab \) gives:
   \[
   -2n - m = -7
   \]

2. The constant term gives:
   \[
   mn = 3
   \]

### Step 3: Solve the system of equations
We have the system:
\[
m + 2n = 7
\]
\[
mn = 3
\]

We can solve for \( m \) in terms of \( n \) from the first equation:
\[
m = 7 - 2n
\]

Substituting this into the second equation:
\[
(7 - 2n)n = 3
\]
\[
7n - 2n^2 = 3
\]
\[
2n^2 - 7n + 3 = 0
\]

### Step 4: Solve the quadratic equation
Solve the quadratic equation using the quadratic formula:
\[
n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For the equation \( 2n^2 - 7n + 3 = 0 \):
\[
n = \frac{7 \pm \sqrt{49 - 24}}{4} = \frac{7 \pm \sqrt{25}}{4} = \frac{7 \pm 5}{4}
\]
\[
n = \frac{12}{4} = 3 \quad \text{or} \quad n = \frac{2}{4} = \frac{1}{2}
\]

### Step 5: Find the corresponding values of \( m \)
- If \( n = 3 \), then \( m = 7 - 2(3) = 1 \).
- If \( n = \frac{1}{2} \), then \( m = 7 - 2\left(\frac{1}{2}\right) = 7 - 1 = 6 \).

### Step 6: Calculate \( m + n \)
- If \( m = 1 \) and \( n = 3 \), then \( m + n = 1 + 3 = 4 \).
- If \( m = 6 \) and \( n = \frac{1}{2} \), then \( m + n = 6 + \frac{1}{2} = \frac{12}{2} + \frac{1}{2} = \frac{13}{2} = 6.5 \).

### Conclusion
One possible value of \( m + n \) is \( \boxed{4} \).

Would you like further details or have any questions?

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Here are 5 related questions:
1. How would you solve a similar problem if the quadratic was \( 3a^2b^2 - 8ab + 5 \)?
2. Can you factorize the expression \( x^2 - 10x + 25 \) using a similar method?
3. How do you derive the quadratic formula used in this solution?
4. What happens if the constant \( mn \) is negative in a similar problem?
5. Can you explain how to complete the square for the quadratic equation derived?

**Tip:** When solving quadratic equations, always check if factoring is possible before applying the quadratic formula.

Learn how to factorize the expression 2(a^2)(b^2) - 7ab + 3 into the form (2ab - m)(ab - n) with detailed step-by-step solution. This problem covers algebraic concepts, including factorization techniques and coefficient comparison. Suitable for high school students in Grades 9-12.

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