From S1:

\((x + m)(x + n) = x^2 + 5x + mn\)

\(x^2+x(m+n)+mn = x^2+5x+mn\)

5x = x(m+n)

x(5-(m+n)) = 0

As x ≠ 0, m+n = 5

Sufficient.

From S2:

Clearly Insufficient.

A is the answer.

We know that \(m\) and \(n\) are integers. The original question: \(m+n=?\)

1) We know that \((x+m)(x+n)=x^2+5x+mn\) and \(x\neq 0\).

\(x^2+mx+nx+mn=x^2+5x+mn\)

\(x(m+n)=5x\)

Since \(x\neq 0\), we can divide both sides of the above equation by \(x\).

\(m+n=5\). Thus, the answer to the original question is a unique value. \(\implies\) Sufficient

2) We know that \(mn=4\) and can test possible cases. If \(m=2\) and \(n=2\), then \(m+n=4\). However, if \(m=4\) and \(n=1\), then \(m+n=5\). Thus, we can't get a unique value to answer the original question. \(\implies\) Insufficient

Answer: A
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Given: m and n are integers

Target question: What is the value of m + n ?

Statement 1: (x + m)(x + n) = x² + 5x + mn and x ≠ 0.
Use FOIL to expand the left side: x² + nx + mx + mn = x² + 5x + mn
Factor the two middle terms: x² + x(n + m) + mn = x² + 5x + mn
At this point, we should see that m+n = 5
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

If you're not convinced, we can take a few more steps.
Take: x² + x(n + m) + mn = x² + 5x + mn
Subtract x² from both sides: x(n + m) + mn = 5x + mn
Subtract mn from both sides: x(n + m) = 5x
Divide both sides by x to get: n + m = 5
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: mn = 4
There are many values of m and n that satisfy statement 2. Here are two:
Case a: m = 4 and n = 1. In this case, the answer to the target question is m + n = 4 + 1 = 5
Case b: m = 2 and n = 2. In this case, the answer to the target question is m + n = 2 + 2 = 4
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer: A

Cheers,
Brent
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Hi All,

We're told that M and N are INTEGERS. We're asked for the value of M+N. This question can be solved with a mix of Algebra and TESTing VALUES.

(1) (X + M)(X + N) = X^2 + 5X + (M)(N) and X ≠ 0.

The equation in Fact 1 should get us thinking about a Quadratic and how we can F.O.I.L. the parentheses and get the given result. Notice that the 'middle' term is "+5X"; this provides a huge clue about how M and N relate to one another. Since the SUM of M and N must be +5, you can stop right here (since the question asks for that exact value). If you didn't immediately recognize that though, you can run a few TESTs and see what happens...

IF....
M=1 and N=4, then (X+1)(X+4) = X^2 +5X + 4 and the answer to the question is 5.
M=2 and N=3, then (X+2)(X+3) = X^2 +5X + 6 and the answer to the question is 5.
M=6 and N= -1, then (X+6)(X-1) = X^2 +5X - 6 and the answer to the question is 5.
Etc.
The answer to the question is ALWAYS 5.
Fact 1 is SUFFICIENT

(2) (M)(N) = 4

With the equation in Fact 2, we can come up with a couple of simple examples...
IF....
M=1 and N=4, then the answer to the question is 5.
M=2 and N=2, then the answer to the question is 4.
Fact 2 is INSUFFICIENT

Final Answer:

GMAT assassins aren't born, they're made,
Rich
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