I think the question is
Find the value of c so that (x-5) is a factor of the polynomial
[tex]p(x) = x^3 + 2x^2 + cx + 10[/tex]
The other factor is going to be some quadratic. We can say a few things about its coefficients but let's start by saying in general it's
[tex]q(x)= ax^2 + bx + k[/tex]
[tex]p(x) = (x-5)q(x)[/tex]
[tex]x^3 + 2x^2 + cx + 10 = (x-5)(ax^2 + bx+k) = ax^3 + (b-5a)x^2 + (k-5b)x - 5k[/tex]
Equating respective coefficients,
[tex]a=1[/tex]
[tex]b-5a = 2[/tex]
[tex]k - 5b = c[/tex]
[tex]-5k = 10[/tex]
so we get
[tex]b = 2 + 5 = 7[/tex]
[tex]k = 10/-5 = -2[/tex]
[tex]c = k - 5b = 2 - 5(7)= -37[/tex]
Answer: -37
Check:
[tex](x^2 + 7x - 2)(x - 5) = x^3 + 2 x^2 - 37 x + 10\quad\checkmark[/tex]
