In the form (ax+b)^2 + c

I’ve got a couple of questions here. The question is: 4x^2 +12x +14. My method was to multiply out (ax+b)^2 + c and solve for a,b,c then sub those back into the form it wants and get (2x+3)^2 +5 which is right. Another method I used was to complete the square in the form a(x+b)^2 + c which gives you 4(x+3/2)^2 +5.

From this I have 2 questions: When you get rid of the fraction in the bracket you multiply by 2 to get 2x+3, but because it’s squared does that mean you’ve multiplied by 4? So the 4 on the outside of the bracket becomes 1 leaving you with again (2x+3)^2 +5? Can someone explain how this part works.

Also what method would you use? Is there one straight up method to get to your answer or do you have to go down another route first like I have.

Thanks

From this I have 2 questions: When you get rid of the fraction in the bracket you multiply by 2 to get 2x+3, but because it’s squared does that mean you’ve multiplied by 4? So the 4 on the outside of the bracket becomes 1 leaving you with again (2x+3)^2 +5? Can someone explain how this part works

Also what method would you use? Is there one straight up method to get to your answer or do you have to go down another route first like I have.

Thanks

I tend to extract the "a" first of all, then complete the square, then multipy through by the "a" to get the desired form.

But, whatever works for you.

Edit: Too slow.

Q1)
(2x+3)^2
= (2(x+3/2))^2
= 2^2 (x+3/2)^2
= 4(x+3/2)^2
So they're the same as you suspect (obviously works in reverse as well).

Q2) I'd complete the square as the terms are a bit more meaningful, but often there are several approaches and your algebraic approach is fine as well.