Use Cauchy's Bound to find an interval \( [a, b] \) which contains all the real zeros of the polynomial \( P(x)=2 x^{4}+8 x^{3}+2 x^{2}+5 x-3 \) : If this is a quiz or exam question, all steps and logic must be shown for full credit. (Use interval notation.)

To use Cauchy's Bound, we consider the polynomial \( P(x) = 2x^4 + 8x^3 + 2x^2 + 5x - 3 \). First, we determine the coefficients \( a_n \) where the leading coefficient is \( a_4 = 2 \), \( a_3 = 8 \), \( a_2 = 2 \), \( a_1 = 5 \), and \( a_0 = -3 \). Next, we find the maximum of \( \frac{|a_i|}{|a_n|} \) for each coefficient \( a_i \) with \( i < n \): 1. \( |a_0|/|a_4| = |-3|/|2| = 3/2 \) 2. \( |a_1|/|a_4| = |5|/|2| = 5/2 \) 3. \( |a_2|/|a_4| = |2|/|2| = 1 \) 4. \( |a_3|/|a_4| = |8|/|2| = 4 \) Next, we find the maximum: \[ M = \max \left( \frac{3}{2}, \frac{5}{2}, 1, 4 \right) = 4 \] Cauchy's Bound states that the real zeros of \( P(x) \) lie within the interval \( [-M, M] \). Therefore, we have: \[ [-4, 4] \] Hence, the interval that contains all the real zeros of the polynomial \( P(x) \) is \( \boxed{[-4, 4]} \).