How to derive (2a(a+1)+1)² from 4a⁴+8a³+8a²+4a+1

[MiddleEast]

Homework Statement

NA

Relevant Equations

NA

Hello,
While following problem solution found this $$ 4a^4 + 8 a^3 + 8 a^2 + 4a + 1 = ( 2a (a+1) + 1 )^2 $$
Trying to figure out how did author do it but failed.
Anyone?

 

[berkeman]

Homework Statement: NA
Relevant Equations: NA

Hello,
While following problem solution found this $$ 4a^4 + 8 a^3 + 8 a^2 + 4a + 1 = ( 2a (a+1) + 1 )^2 $$
Trying to figure out how did author do it but failed.
Anyone?

Well, what happens when you multiply out the RHS? Can you show those steps to see how close you get to the LHS?

 

[MiddleEast]

Thanks for quick reply. It is simple to start with R.H.S to L.H.S.
As per solution, they move from L.H.S to R.H.S, thats my question how to play with it?
Tried to get different common factors over and over, no success. It is supposed to move from L.H.S to R.H.S.

 

[berkeman]

Just factor the LHS then. Do it in a first step to get <something> squared, and then look to simplify what is inside the squaring parenthesis...

 

[vela]

Homework Statement: NA
Relevant Equations: NA

Hello,
While following problem solution found this $$ 4a^4 + 8 a^3 + 8 a^2 + 4a + 1 = ( 2a (a+1) + 1 )^2 $$
Trying to figure out how did author do it but failed.
Anyone?

Try rewriting $8a^2$ as $4a^2+4a^2$. I'm not sure what would motivate that other than trying to write the first three terms as a perfect square.

 

[PeroK]

Homework Statement: NA
Relevant Equations: NA

Hello,
While following problem solution found this $$ 4a^4 + 8 a^3 + 8 a^2 + 4a + 1 = ( 2a (a+1) + 1 )^2 $$
Trying to figure out how did author do it but failed.
Anyone?

If the RHS is a perfect square, then it must be of the form $(\alpha a^2 + \beta a +1)^2$.

Now, solve for $\alpha, \beta$.