Methods/approach to simplify $k^2(k+1)^2 + 4(k+1)^3$
Hi guys are there any other Methods/approach to simplify $k^2(k+1)^2 + 4(k+1)^3$ instead of expanding everything out and getting a quartic and cubic?? I expanded the entire thing out and got a quartic and cubic which both i have used factor theorem and long division. In the end i have got $(k+1)^2 \cdot (k+2)^2$ which is the answer, but this approach took too long to solve. So my question is are there any more simpler approach so can i do this quicker?
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$\begingroup$Do you really mean $k^2(k+1)^2\cdot 4(k+1)^3=4k^2(k+1)^5$ which should not be simplyfied further, because it does not get simpler then that.
It always depends on what you want to do with your term.
If you mean $k^2(k+1)^2+4(k+1)^3$ you can factor out $(k+1)^2$ and get:
$(k+1)^2(k^2+4k+4)=(k+1)^2(k+2)^2$
Where we used the binomial formula, to see $k^2+4k+4=(k+2)^2$
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