Solve the following systems of equations: p+r = -10 (ii) p+4r = -10

$$
p(-1) = (-1)^3 + (-1)^2 + (-1) + 1
$$

$$
p(-1) = -1 + 1 - 1 + 1 = 0
$$
Since $p(-1) = 0$, $(x+1)$ is a factor of $x^3 + x^2 + x + 1$.

$$
p(-1) = (-1)^4 + (-1)^3 + (-1)^2 + (-1) + 1
$$

$$
p(-1) = 1 - 1 + 1 - 1 + 1 = 1
$$
Since $p(-1) \neq 0$, $(x+1)$ is not a factor of $x^4 + x^3 + x^2 + x + 1$.

$$
p(-1) = (-1)^4 + 3(-1)^3 + 3(-1)^2 + (-1) + 1
$$

$$
p(-1) = 1 + 3(-1) + 3(1) - 1 + 1
$$
$$
p(-1) = 1 - 3 + 3 - 1 + 1 = 1
$$
Since $p(-1) \neq 0$, $(x+1)$ is not a factor of $x^4 + 3x^3 + 3x^2 + x + 1$.

$$
p(-1) = (-1)^3 - (-1)^2 - (2+\sqrt{2})(-1) + \sqrt{2}
$$

$$
p(-1) = -1 - 1 + (2+\sqrt{2}) + \sqrt{2}
$$
$$
p(-1) = -2 + 2 + \sqrt{2} + \sqrt{2} = 2\sqrt{2}
$$
Since $p(-1) \neq 0$, $(x+1)$ is not a factor of $x^3 - x^2 - (2+\sqrt{2})x + \sqrt{2}$.

$$
p(-1) = 2(-1)^3 + (-1)^2 - 2(-1) - 1
$$

$$
p(-1) = 2(-1) + 1 - (-2) - 1
$$
$$
p(-1) = -2 + 1 + 2 - 1 = 0
$$
Since $p(-1) = 0$, $g(x)$ is a factor of $p(x)$.

$$
p(-2) = (-2)^3 + 3(-2)^2 + 3(-2) + 1
$$

$$
p(-2) = -8 + 3(4) - 6 + 1
$$
$$
p(-2) = -8 + 12 - 6 + 1
$$
$$
p(-2) = 4 - 6 + 1 = -1
$$
Since $p(-2) \neq 0$, $g(x)$ is not a factor of $p(x)$.

$$
p(3) = (3)^3 - 4(3)^2 + 3 + 6
$$

$$
p(3) = 27 - 4(9) + 3 + 6
$$
$$
p(3) = 27 - 36 + 3 + 6
$$
$$
p(3) = -9 + 3 + 6 = 0
$$
Since $p(3) = 0$, $g(x)$ is a factor of $p(x)$.