How to factor 10k^2+35k+15 completely?
I would start with the easy: factor five out.
5(5k^2 +7x +3)
Now the three tells you that the factors are probably 3,1 and the five tells you that the factors are probably 5,1
5(5k+3)(k+1) wrong, wont work
and 5(5k+1)(k+3) wont work. So the easy is out. Next, the quadratic equation...
k= -b +- sqrt (b^2 -4ac) /2a
k= -7 +- sqrt (49 - 60) /10
Notice the serd: it is negative, so the factors are in the the imaginary plane. You cannot factor the problem in the real number system.