Factoring perfect squares: negative common factor

we need to factor negative 40 squared minus 12t minus 9 and a good place to start just say well are there any common factors for all of these terms when you look at them well these first two are divisible by 4 these last two are divisible by 3 but not all of them are divisible by any one number well but you could factor out a negative one but even if you factor out a negative one so you say this is the same thing as negative 1 times positive 4 T squared plus twelve T plus 9 you still end up with a non-one coefficient out here and on the second degree term on the T squared term so you might want to immediately start grouping this and if you did group if you did factor it by grouping it would work you would get the right answer but there is something you might be able to or there is something about this equation that might pop at it out of you that might make it a little bit simpler to solve and to understand that let's take a little bit of a of a of a break here on the right hand side and just think about what happens if you take a plus B times a plus B if you just have you know a binomial squared well you have a times a which is a squared then you have a times that B which is plus a B then you have B times a which is the same thing as a B and then you have B times B or you have B squared and so if you add these middle two terms right here you're left with a squared plus 2 a B plus B squared this is the square of a binomial now does this right here does 40 squared plus 12t plus 9 fit this pattern well the 40 squared is a squared so if this right here is a squared if that is a squared right there then what does a have to be if this is a squared then a would be equal to the square root of this it would be 2t and if this is B squared let me do that in a different color if if this right here is B squared if the 9 is B squared right there then that means that B is equal to 3 it's equal to the positive square root of the nine now this number right here and actually doesn't have to just be equal to three it might have been negative 3 as well it could be plus or minus 3 but this number here is it two times a B right that's the middle term that we care about is it two times a B well if we multiply 2t times three we get 60 and then we multiply that times two you get twelve T this right here twelve T is equal to two times 2t times three it is two times a B and if this was a negative three we would look to see if this was a negative twelve but this does work for a positive three so this does fit the pattern of a perfect squared this is a this is a special type of ways you could do this is a square of a binomial so if you wanted to factor this the stuff on the inside you still have that negative one out there the 4t squared plus 12t plus nine you could immediately say well that's going to be a plus B times a plus B or or two T to T plus three two T plus three times 2t plus three or you could just say it's 2t plus three squared it fits this pattern and of course you can't forget about this negative one out here you could have also solved it by grouping but this might be a quicker thing to recognize this is a number squared that's another number squared if you take each of those numbers that you're squaring take their product and multiply it by two you have that right there so this is a perfect square