Multiply the polynomials.(x+4)(3x2−2x+1)\left(x+4\right)\left(3x^2-2x+1\right)(x+4)(3x2−2x+1)

3 x 3 + 10 x 2 − 7 x + 4 3x^3+10x^2-7x+4 3 x 3 + 10 x 2 − 7 x + 4

Multiply the polynomials. ( x + 4 ) ( 3 x 2 − 2 x + 1 ) \left(x+4\right)\left(3x^2-2x+1\right) ( x + 4 ) ( 3 x 2 − 2 x + 1 )

Alright, everyone. Hopefully, you got a chance to check out this practice problem here. So we have to multiply these polynomials. I've got x plus 4, and I've got 3x2minus2xplus1. So how do I multiply them? One thing you might be thinking is, can I just foil these? Well, remember, foiling is gonna work for when you have 2 term time, 3 term expressions because if you try to foil, what's gonna happen is you're gonna multiply all these things, but you're gonna leave one of the terms out. You're gonna do inner terms and then finally last terms. This negative two x never gets multiplied by anything, so you can't do that. So how do we do this? We basically just split it into a distribution problem. We're gonna take the shortest expression the x plus 4 and we're gonna split that out. So how does this work? I'm gonna take the x over here and then the 4 over here, and I'm gonna put parenthesis. And then basically, all I'm gonna do here is I'm just gonna rewrite this expression. So this is x times 3x2-2x+1 plus 4 parenthesis 3x2-2x plus 1. So you just break it up into 2 parts, and now you can basically just do distributive property for both of these problems. So let's take a look at how that happens. I'm gonna take the x, distribute it into everything inside over here, and this becomes 3x3. The those so that's what this term becomes. And I've got the negative 2x. If I multiply by another x, it just becomes, negative 2x squared. And then if I take the 1 and just multiply by x, it becomes plus x. Now over here, what happens is I take the 4, and this becomes 4 times 3x squared. So 4 times 3x squared becomes 12x squared, then the 4 into the negative 2 becomes negative 8x, and the 4 into the one just becomes 4. So now what I can do is this is the distributive property. Notice how I got 123, 4, 5, and 6 terms, and that's exactly how many I should get because, again, I'm multiplying a 2 term by 3 term expression. Okay. So now I just have to go and had combine and that's my last step. So I just have to look for any anything that's, you know, a 3 that that's x cubed. I see a 3 x cubed over here, but that's the only power of 3 that I see. Here, I've got a negative 2 x squared and a 12 x squared that's positive. So those are my x squareds, and I look look at my x's. I've got an x and an in negative 8x, and then finally, I've got a just a constant of 4. Alright? So if you go ahead and combine everything, what happens is the 3x3 doesn't combine with anything, but then I've gotta do the x squared, negative 2, and then 12x squared just becomes 10x squared. The x and negative 8x becomes negative 7x, and then the 4 just drops down over here. So this is my simplified expression. Can't simplify anything further because of a combine all the like terms, and that's how to multiply these 2 polynomials. That's your final answer. Hopefully you got that right. Let me know if you have any questions. Thanks for watching.