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the quadratic expressions 4x squared plus 12x plus 9 and 4x squared minus 9 share a common binomial factor what binomial factor do they share and I encourage you to pause the video see if you can figure it out so let's do this by taking each of these expressions and trying to factor them into binomials and then see if they share a common binomial factor I guess they do share wanted to figure out which ones are the which one they actually share so let's focus first on this 4x squared plus 12x plus 9 so the first thing that might jump out at you is well let's see I have a 4 here this coefficient on the x squared term that's a perfect square I could write the entire 4x squared term I could rewrite that as 2x to the 2nd power then out here I have a constant term the 9 that also is a perfect square I could rewrite that as 3 squared and you could say well gee what you know does this does this fit the pattern of a perfect square polynomial in order for it to fit the pattern of a perfect square polynomial the coefficient here on the X term would need to be 2 times the product of this 2 and this 3 and it is indeed 2 times the product of 2 and 3 is 2 times 6 so we could write this we could write this part right over here as 2 times 2 times that 2 times 3 times that 3 X X and then of course we have to add these three things together so plus plus and so just like that we can recognize hey this is a this is a perfect square polynomial right over here and if what I'm what I'm saying right now sounds like a little bit of Voodoo I encourage you to watch some of the videos on perfect square trinomials perfect square problem is some of the last few videos in this progression so this thing can be can be re-written as the same thing as 2 X plus 3 2 X plus 3 squared 2x plus 3 squared once again because it's of the form you have the entire 2x squared here you have the entire you have the 3 squared here and then this middle term is 2 times the product of these two terms right over here and so it definitely it definitely fit the pattern so there you have it we factored the first expression and now let's try to factor the second expression and immediately when you see this one it looks like it's a difference of squares so this one right over there looks like a difference of squares to me this we can rewrite as 2 x squared minus - pick a nice color - 3 squared so minus 3 squared this is a difference of squares we've seen multiple times how to factor a difference of squares if this is once again looks foreign to you I encourage you to watch those videos that we explain how that works and why it works but this is going to be when you have something of the form a squared minus B squared it's going to be equal to a plus B times a minus B so this is going to be equal to let me just put the two binomials right over here so it's gonna be a plus B times a minus B so this is going to be 2x plus 3 times 2x minus 3 so the 2x plus 3 times 2x minus 3 and so what is their common binomial factor well they both involve when you when you factor them out they both have a binomial factor of 2x plus 3 this one right over here we can rewrite if we want we could rewrite it as 2x plus 3 times 2x plus 3 that might have been somewhat obvious to you already so 2x 2x and then you have plus 3 plus 3 these two are equivalent and so you see we see that we share we in both in both of these we share at least one or we share exactly one 2x plus 3 so that's the binomial factor that they share 2x plus 3