Resposta:

[tex]\textsf{Leia abaixo}[/tex]

Explicação passo a passo:

[tex]\mathsf{p(x) = x^4 - x^3 - (p + 10)x^2 -(q - p)x + 30}[/tex]

[tex]\mathsf{r_1 = -2}[/tex]

[tex]\mathsf{0 = (-2)^4 - (-2)^3 - (p + 10)(-2)^2 -(q - p)(-2) + 30}[/tex]

[tex]\mathsf{0 = (16) - (-8) - (p + 10)(4) -(q - p)(-2) + 30}[/tex]

[tex]\mathsf{0 = (16) - (-8) - (4p + 40) -(-2q + 2p) + 30}[/tex]

[tex]\mathsf{0 = 16 + 8 - 4p - 40 + 2q - 2p + 30}[/tex]

[tex]\mathsf{6p - 2q = 14}[/tex]

[tex]\mathsf{3p - q = 7}[/tex]

[tex]\mathsf{r_2 = 1}[/tex]

[tex]\mathsf{0 = (1)^4 - (1)^3 - (p + 10)(1)^2 -(q - p)(1) + 30}[/tex]

[tex]\mathsf{0 = 1 - 1 - p - 10 - q + p + 30}[/tex]

[tex]\mathsf{q = 20}[/tex]

[tex]\mathsf{3p - 20 = 7}[/tex]

[tex]\mathsf{3p = 27}[/tex]

[tex]\mathsf{p = 9}[/tex]

[tex]\mathsf{p(x) = x^4 - x^3 - (9 + 10)x^2 -(20 - 9)x + 30}[/tex]

[tex]\mathsf{p(x) = x^4 - x^3 - 19x^2 - 11x + 30}[/tex]

[tex]\mathsf{\dfrac{x^4 - x^3 - 19x^2 - 11x + 30}{(x - 1).(x + 2)}}[/tex]

[tex]\mathsf{\dfrac{x^4 - x^3 - 19x^2 - 11x + 30}{x^2 + x - 2} = x^2 - 2x - 15}[/tex]

[tex]\mathsf{x^2 - 2x - 15 = (x + 3).(x - 5)}[/tex]

[tex]\boxed{\boxed{\mathsf{S = \{-2;1-3;5\}}}}[/tex]