Determine the value(s) of \( m \) for which the roots of the equation are non-real. Given that \( f(x)=x^{2}-p x+8+2 p \) has two equal root and \( p<0 \), determine the coodinates of the turning point, \( h(x)=f(x)-3 \).

To find the values of \( m \) for which the roots of the equation are non-real, we need to look at the discriminant of the quadratic equation. The equation is given as \( f(x) = x^2 - p x + (8 + 2p) \). The discriminant \( \Delta \) is given by: \[ \Delta = b^2 - 4ac = (-p)^2 - 4(1)(8 + 2p) = p^2 - 32 - 8p \] Setting the discriminant less than zero for non-real roots: \[ p^2 - 8p - 32 < 0 \] Now, we can analyze the quadratic \( p^2 - 8p - 32 \) using the quadratic formula to find the critical points: \[ p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{8 \pm \sqrt{(-8)^2 - 4(1)(-32)}}{2(1)} \] \[ = \frac{8 \pm \sqrt{64 + 128}}{2} = \frac{8 \pm 12}{2} \] This gives us \( p = 10 \) and \( p = -2 \). Since \( p < 0 \), we need to find what part of the interval \( (-\infty, -2) \) gives us non-real roots. The roots of the original quadratic are non-real when \( -2 < p < 10 \), taking into the account the condition \( p < 0\), we conclude that non-real roots exist for \( -2 < p < 0 \). Now, for the function \( h(x) = f(x) - 3 \): We will calculate the turning point of \( f(x) \) first. The vertex (or turning point) of the quadratic is located at: \[ x = -\frac{b}{2a} = -\frac{-p}{2} = \frac{p}{2} \] Substituting \( x = \frac{p}{2} \) back into \( f(x) \): \[ f\left(\frac{p}{2}\right) = \left( \frac{p}{2} \right)^2 - p\left( \frac{p}{2} \right) + (8 + 2p) \] \[ = \frac{p^2}{4} - \frac{p^2}{2} + 8 + 2p = \frac{p^2}{4} - \frac{2p^2}{4} + 8 + 2p = -\frac{p^2}{4} + 8 + 2p \] Now, substituting into \( h(x) = f(x) - 3 \): The y-coordinate of the turning point for \( h(x) \) is: \[ h\left(\frac{p}{2}\right) = f\left(\frac{p}{2}\right) - 3 \] \[ = -\frac{p^2}{4} + 8 + 2p - 3 = -\frac{p^2}{4} + 2p + 5 \] Thus, the coordinates of the turning point of \( h(x) \) are: \[ \left(\frac{p}{2}, -\frac{p^2}{4} + 2p + 5\right) \] So the coordinates of the turning point can be explicitly calculated if we plug in any specific value of \( p \) which satisfies \( p < 0 \).