Question 153988
There is only one error in your working. That happened when you expand 
{{{6(x + 25)*25}}}
Its expanded form is:
{{{6(x + 25)*25}}}
={{{6*25(x+25)}}}
={{{6*25x +6*25*25}}}  (multiply each terms in the parentheses by 6*25)
={{{150x+6*25*25}}}

So
{{{25(6x)=6*(x+25)*25-x(x+25)}}}
becomes:
{{{150x=150x+6*25*25-x^2-25x}}}
Combining like terms, we have:
{{{x^2+25x-6*25*25=0}}}
To factor the left side, note that -6*25*25 =3*25*(-2*25)=75*(-50).
As 75 + (-50) = 25, so {{{x^2+25x-6*25*25=(x+75)(x-50)}}}
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Also note that there is an typing error in the equation:
25(6x) = 6(x+25)(25) + x(x+25)
The sign before the term x(x+25) should be "-".