Question 598942
We start by dividing {{{25x^3/(-5x^2)=-5x}}} to get the first term of the quotient:
{{{drawing(300,100,-2.5,3.5,-1.5,1.5,
line(-0.1,0.1,4,0.1), line(-0.1,0.1,-0.1,-0.8),
locate(0,0.75,-5x),locate(-2.5,0,-5x^2-3x-2),
locate(0,0,25x^3), locate(0.8,-0.25,"+"),locate(1,0,40x^2),
locate(1.8,-0.25,"+"),locate(2,-0.25,25x),
locate(2.8,-0.25,"+"),locate(3,-0.25,10)
)}}}
Then, we multiply that first term {{{-5x}}} times divisor {{{-5x^2-3x-2}}} and subtract the product
{{{-5x(-5x^2-3x-2)=25x^3+15x^2+10x}}} from {{{25x^3+40x^2+25x+10}}}
to get {{{25x^3+40x^2+25x+10-(25x^3+15x^2+10x)=25x^2+15x+10}}}
{{{drawing(300,150,-2.5,3.5,-3.2,1.3,
line(-0.1,0.1,4,0.1), line(-0.1,0.1,-0.1,-0.8),
locate(0,0.75,-5x),locate(-2.5,0,-5x^2-3x-2),
locate(0,0,25x^3), locate(0.8,-0.25,"+"),locate(1,0,40x^2),
locate(1.8,-0.25,"+"),locate(2,-0.25,25x),
locate(2.8,-0.25,"+"),locate(3,-0.25,10),
locate(1,0.75,-5),
locate(0,-1,25x^3), locate(0.8,-0.25,"+"),locate(1,-1,15x^2),
locate(1.8,-1.25,"+"),locate(2,-1.25,10x),
blue(line(-0.1,-1.9,4,-1.9)),
locate(1,-2,25x^2),
locate(1.8,-2.25,"+"),locate(2,-2.25,15x),
locate(2.8,-2.25,"+"),locate(3,-2.25,10),
blue(line(-0.3,-1.1,-0.1,-1.1))
)}}}
Now we have to divide that {{{25x^2+15x+10}}} difference by  {{{-5x^2-3x-2}}}.
To do that, the first step is dividing {{{25x^2/(-5x^2)=-5}}} to get the next (second) term of the quotient:
{{{drawing(300,150,-2.5,3.5,-3.2,1.3,
line(-0.1,0.1,4,0.1), line(-0.1,0.1,-0.1,-0.8),
locate(0,0.75,-5x),locate(-2.5,0,-5x^2-3x-2),
locate(0,0,25x^3), locate(0.8,-0.25,"+"),locate(1,0,40x^2),
locate(1.8,-0.25,"+"),locate(2,-0.25,25x),
locate(2.8,-0.25,"+"),locate(3,-0.25,10),
locate(1,0.75,-5),
locate(0,-1,25x^3), locate(0.8,-0.25,"+"),locate(1,-1,15x^2),
locate(1.8,-1.25,"+"),locate(2,-1.25,10x),
blue(line(-0.1,-1.9,4,-1.9)),
locate(1,-2,25x^2),
locate(1.8,-2.25,"+"),locate(2,-2.25,15x),
locate(2.8,-2.25,"+"),locate(3,-2.25,10),
locate(1,0.75,-5),blue(line(-0.3,-1.1,-0.1,-1.1))
)}}}
Then, we multiply that second term {{{-5}}} times divisor {{{-5x^2-3x-2}}} and subtract the product
{{{-5(-5x^2-3x-2)=25x^2+15x+10}}} from {{{25x^2+15x+10}}}
to get the remainder, which happens to be zero:
{{{25x^2+15x+10-(25x^2+15x+10)=0}}}
{{{drawing(300,200,-2.5,3.5,-5,1,
line(-0.1,0.1,4,0.1), line(-0.1,0.1,-0.1,-0.8),
locate(0,0.75,-5x),locate(-2.5,0,-5x^2-3x-2),
locate(0,0,25x^3), locate(0.8,-0.25,"+"),locate(1,0,40x^2),
locate(1.8,-0.25,"+"),locate(2,-0.25,25x),
locate(2.8,-0.25,"+"),locate(3,-0.25,10),
locate(1,0.75,-5),
locate(0,-1,25x^3), locate(0.8,-0.25,"+"),locate(1,-1,15x^2),
locate(1.8,-1.25,"+"),locate(2,-1.25,10x),
blue(line(-0.1,-1.9,4,-1.9)),
locate(1,-2,25x^2),
locate(1.8,-2.25,"+"),locate(2,-2.25,15x),
locate(2.8,-2.25,"+"),locate(3,-2.25,10),
locate(1,0.75,-5),
locate(1,-3,25x^2),
locate(1.8,-3.25,"+"),locate(2,-3.25,15x),
locate(2.8,-3.25,"+"),locate(3,-3.25,10),
locate(3.1,-4.25,0), blue(line(0.8,-4,4,-4)),
blue(line(-0.3,-1.1,-0.1,-1.1)),blue(line(0.8,-3.1,1,-3.1))
)}}}