We are given #y=x^3+4x^2-x-4#. Because the highest exponent is #3#, we know there are #3# solutions.
With anything larger than a quadratic (#ax^2+bx+c#), I like to use synthetic division. To use synthetic division, the first thing we must do is find the divisor. We can do that by guessing-and-checking, or we can graph the equation and use the roots (x-intercepts) as our divisor. Let's do that right now:
graph{y=x^3+4x^2-x-4}
Lucky for us, we can see all three solutions. I'm still going to use synthetic division just to show you how it works:
#color(white)(-1.)|##color(white)(.)1##color(white)(...)4##color(white)(.)-1##color(white)(.)-4#
#-1color(white)(.)|#_#-1#__#-3#__#4#_
#color(white)(.........)1color(white)(...)3color(white)(..)-4color(white)(....)0#
This can be rewritten as #x^2+3x-4#, which can be factored to #(x-1)(x+4)#.
So, we have #(x+4)(x-1)(x+1)#! And that's what the graph shows too. Nice work!
