Given that y=(x+p)(x^2-mx+n), calculate the value of x for dy=0?

For the beginning, let's differentiate the
given function.

dy=(x+p)(x^2-mx+n)dx

Since
the function is a product, we'll apply the product rule, when differentiating a
product.

(f*g)' = f'*g +
f*g'

We'll differentiate, to the right side, with respect
to x:

[ (x+p)(x^2-mx+n) ]' = (x+p)' * (x^2-mx+n) + (x+p) *
(x^2-mx+n)'

[ (x+p)(x^2-mx+n) ]' = x * (x^2-mx+n) + (x+p) *
(2x-m)

We'll remove the
brackets:

[ (x+p)(x^2-mx+n) ]' = x^3 - mx^2 + nx + 2x^2 -
mx + 2px - mp

Now, we'll put dy =
0

We'll substitute the expression for
dy:

x^3 - mx^2 + nx + 2x^2 - mx + 2px - mp =
0

We'll combine like
terms:

x^3 + x^2*(-m+2) + x*(n-m+2p) - mp =
0

If we'll plug in values for the m, n, p, we'll calculate
the values of x for dy = 0.