If a x plus two times b x plus seven is equal to 15 x squared plus c x plus 14 for all values of x, and a plus b is equal to eight, what are the two possible values for C? All right. So what you'd want to do here is basically, instead of having two unknowns, you want to have just one. And how can we do that? Well, we need another relationship that relates a envy. And how can we get that?

All right? Well, we can use this first equation here. And if we multiply this throughout, let's see what we get. We will get a B x squared plus seven, eight x plus two, b x plus 14 is equal to 15 x squared plus c x plus four teen. All right, so what you see here is the coefficient in front of the x squared on the right which happens to be 15 is also the same coefficient. That is in front of the expert on the left, which happens to be a B.

So now we have another relationship. So this relationship is a time speed is equal to 15. And we have the earlier relationship of A plus B is equal to eight, okay? So we can actually use this relationship to say that v is equal to eight minus eight. And if we plug in that value for B in here, then we'll get what, then we'll get eight times eight minus a is equal to 15. And if we foil that out, we will get a squared minus a plus 15 is equal to zero.

Okay. So from there, what we can do is we can factor this and what you'll get is a minus three times a minus Make sure you know how to get from this to this very, very important. Okay, so now that we're here, which is equal to zero, we can say that, well, we have a, a can either equal to three, or it can be equal to five. And if it's three, that means five v must be five, okay? And if it's a is equal to five, then B must be three. Okay?

And why is that? Remember that one of the relationships is a times b is equal to 15. So three times five is equal to 15, or five times three is equal to 15. All right? So what we do now is we take this set of numbers, these values for A and B, and we plug them back all the way into, I can get it here. You can plug it all the way back into this equation right here and see what that gives us.

And if you do that, what you'll see is the phone For these sets of numbers, you will get 15 x squared plus two x 31 x plus 14. And look at that the value in front of the x 31 is our C, because that was the earlier coefficient in front of our x, which is C, right here and right there. Alright, so one of our answers, one of the possible values for C is 31. And you can actually just stop right there, because the only answer choice that has a 31 is D. But if you just want to make sure and if you have a little bit more time, then you can actually plug in these second set of values for A and B, and you will get 15 x squared plus 41 x plus 14 and this is your second possible value for C which is also the other one right there.

So your answers is d