The reots of the equation \( f(x)=0 \) is \( x=\frac{4 \pm \sqrt{6-4 m /-m+5)}}{2 m} \) Determine the valaes of \( m \) for which the roots are non-real

We are given that the roots of f(x) = 0 are   x = [4 ± √( (6 – 4m)/(–m + 5) )] / (2m). For the roots to be non‐real the expression under the square root (the discriminant) must be negative. In other words, we require   (6 – 4m)/(–m + 5) < 0. Step 1. Rewrite the inequality in a slightly simpler form. Notice that   –m + 5 = 5 – m. So the inequality becomes   (6 – 4m)/(5 – m) < 0. Step 2. Find the critical values by setting the numerator and denominator equal to zero. • Numerator: 6 – 4m = 0 ⇒ 4m = 6 ⇒ m = 3/2. • Denominator: 5 – m = 0 ⇒ m = 5. Also note that m = 5 must be excluded since it would make the denominator zero (and would also make the expression under the square root undefined). Additionally, since m appears as a coefficient in the denominator of the entire root expression (x = …/(2m)), one might exclude m = 0 to maintain a quadratic formula—but our focus here is determining when the discriminant is negative. Step 3. Determine the sign of the expression (6 – 4m)/(5 – m) in the intervals determined by m = 3/2 and m = 5. The intervals to test are:  (a) m < 3/2,  (b) 3/2 < m < 5,  (c) m > 5. Test a value in each region: (a) For m = 0 (which is less than 3/2):   Numerator: 6 – 4(0) = 6  (positive).   Denominator: 5 – 0 = 5  (positive).   So (6 – 4m)/(5 – m) > 0. (b) For m = 2 (which lies between 3/2 and 5):   Numerator: 6 – 4(2) = 6 – 8 = –2  (negative).   Denominator: 5 – 2 = 3  (positive).   So (6 – 4m)/(5 – m) < 0. (c) For m = 6 (which is greater than 5):   Numerator: 6 – 4(6) = 6 – 24 = –18  (negative).   Denominator: 5 – 6 = –1  (negative).   So (6 – 4m)/(5 – m) > 0 (since a negative divided by a negative is positive). Step 4. Conclusion The expression (6 – 4m)/(5 – m) is negative only on the interval   3/2 < m < 5. Thus, the roots of the equation are non‐real if and only if m is in the open interval (3/2, 5). Answer: The roots are non‐real for all m satisfying 3/2 < m < 5.