Question 971204
Please refer now to this request question posting:
<a href="http://www.algebra.com/algebra/homework/quadratic/Quadratic_Equations.faq.question.971209.html">http://www.algebra.com/algebra/homework/quadratic/Quadratic_Equations.faq.question.971209.html</a>


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Description is poorly written.  m for man s for son;
The inequality shown does not make sense.  {{{x<2x^2}}}, is what?  
{{{m<x<2x^2}}} and {{{s=x}}}?
This could be {{{m<2x^2}}}.


Let 4x years pass; then 
{{{2(m+4x)=(s+4x)}}}


The only understanding is a system
{{{system(m<2x^2,2(m+4x)=s+4x)}}}.


{{{2m+8x=s+4x}}}
{{{2m-s=-8x+4x}}}
{{{2m-s=-4x}}}
{{{2m=s-4x}}}
{{{m=(s-4x)/2}}}


Use m in the inequality.
{{{(s-4x)/2<2x^2}}}
{{{s-4x<4x^2}}}
{{{s-4x-4x^2<0}}}
{{{-s+4x+4x^2>0}}}
{{{4x^2+4x-s>0}}}, a parabola with a minimum for a vertex.  The inequality will be true to the left of the lesser root and to the right of the greater root.  The polynomial must have two real roots.

Look at the discriminant.
{{{4^2-4*4*(-s)>0}}}, because the discriminant must be positive.
{{{16+16s>0}}}
{{{1+s>0}}}
{{{1>-s}}}
{{{-1<s}}}------------which seems to indicate that the son is ONE YEAR BEFORE BEING BORN.


Write the inequality using this value for s.
{{{highlight_green(4x^2+4x-(-1)>0)}}}
{{{4x^2+4x+1>0}}}
Now what are the roots for this polynomial?
{{{x=(-4+- sqrt(16-4*4(-1)))/8}}}
{{{x=(-4+- sqrt(32))/8}}}
{{{x=(-4+- sqrt(2^5))/8}}}
{{{x=(-4+- 16sqrt(2))/8}}}
{{{x=(-1/2+- 2*sqrt(2))}}}
We must use the positive, or PLUS form for x.
{{{highlight(x>2sqrt(2)-1/2)}}}, in order to satisfy the inequality.


Look carefully at the problem description as it is given to you and write the exact problem description, word for word to be sure you did not mis-state it in the help request.  What is shown in your request here seems overly complicated but appeared finally solvable.