Question 1172699
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Such problem may perplex an average person.


No, be sure, I know that I can solve a quadratic equation and find the roots,
but the problem is not about solving a quadratic equation.


I know that I can pick up coefficients making trials and errors  manipulating with coefficients -
but the problem is not about it.


    +-------------------------------------------------------------+
    |    The problem is about solving it quickly and mentally,    |
    |                without using heavy artillery . . .          |
    +-------------------------------------------------------------+


OK, then I divide the given polynomial MENTALLY by x^2 and get  {{{3}}} + {{{4/x}}} + {{{1/x^2}}}.


I then replace  {{{1/x}}}  by new variable "u"  (MENTALLY)  and get  new polynomial  {{{u^2 + 4u + 3}}}

with the leading coefficient of 1.



For this polynomial, every person (including me) can guess its factoring in 1-2-3 seconds: it is  {{{u^2 + 4u + 3}}} = (u+1)*(u+3).


Now I take step back making replacement  u = {{{1/x}}}  MENTALLY.


It gives me factoring of the original polynomial as


    3x^2 + 4x + 1 = (x+1)*(3x+1).


and leads to the final answer.
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This description is long &nbsp;(and may seem to be long), &nbsp;but the human's thought runs much faster and gives the answer in seconds.


So, &nbsp;the problem can be solved in this way in &nbsp;5 &nbsp;seconds &nbsp;MENTALLY, &nbsp;without using paper and pencil or computer.



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;OK, &nbsp;may be you need to write one line on the paper . . . 



The way is simply to convert the given quadratic polynomial into other quadratic polynomial with the leading coefficient of &nbsp;1.


Then the guessing is &nbsp;2 &nbsp;seconds and returning back is another &nbsp;2 &nbsp;seconds - - - and everything can be done in 5 seconds.
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