MHB Simplifying the Product Rule for Derivatives

[lastochka]

Hello,

I have this exercise that I can't get the right answer. I have to find derivative of

g(x)= (4${x}^{2}$-2x+1)${e}^{x}$

So, what is did is

g$^{\prime}$=(8x-2)${e}^{x}$+(4${x}^{2}$-2x+1)${e}^{x}$

My Prof said it is wrong... I am not sure if I have to multiply the brackets or what I did is completely wrong.
Can someone, please, check it for me?
Thank you!

 

[MarkFL]

Technically, it is not wrong, however, it can be simplified by factoring...what do you get when you fully factor?

 

[lastochka]

Thank you for answering! I am not sure how to simplify it by factoring...

 

[MarkFL]

Thank you for answering! I am not sure how to simplify it by factoring...

There is a factor common to both terms in your result...can you spot it?

 

[lastochka]

Yes, I see that, but will it make it simplified...
Here is what I have
${e}^{x}$(4${x}^{2}$-6x-1)
Is that it?
Thank you for helping!

 

[MarkFL]

You are close...here's what I get:

$$g'(x)=(8x-2)e^x+\left(4x^2-2x+1\right)e^x=e^x\left(8x-2+4x^2-2x+1\right)=e^x\left(4x^2+6x-1\right)$$

 

[lastochka]

Oh, sorry I mistype, it is plus 6x not minus...
Thank you so much for helping!

 

[MarkFL]

Good deal! Wouldn't you say that is simpler than the unfactored version? :D