Which of the following alkenes would produce two products when they undergo an addition reaction with hydrogen bromide?
Well as mithoron said.its about symmetry.Let's see how but-1-ene comes out to be the answer

On addition of HBr markownikov rule is followed but lets draw both product.

1.)CH2=CH2 + HBr--->Br-CH2-CH3 + CH3-CH2-Br  (both products are 1-bromoethane)

2)CH2=CH-CH2-CH3 + HBr--->Br-CH2-CH2-CH2-CH3 + CH3-BrCH-CH2-CH3 (notice the                name of both the product. 1st one is 1-bromobutane. 2nd one is 2-bromobutane)

3)CH3-CH=CH-CH3 + HBr---->CH3-BrCH-CH2-CH3 + CH3-CH2-BrCH-CH3 (both products are same i.e both are 2-bromobutane)

4)CH3-CH2-CH=CH-CH2-CH3 + HBr--->CH3-CH2-BrCH-CH2-CH2-CH3 +
                    CH3-CH2-CH2-BrCH-CH2-CH3 (here also both the compounds are same i.e both are 3-bromohexane)

So what distinguishes the 2nd one from these three is that the second molecule  is unsymmetrical.You can find its unsymmetrical by looking at the no:of carbon to both side of double bond.If they are equal then molecule is symmetrical.