4m³ + m² - m + 5 = 0

The candidates for rational solutions 
are ± the fractions whose

1. numerator is a divisor of the absolute value of the constant term +5

and whose

2. denominator is a divisor of the absolute value of the leading coefficient 4.

the candidates for solutions are.

The divisors of 5 are 5 and 1
The divisors of 4 are 4,2, and 1

So the candidates for rational zeros are

 = ±1.25
 = ±2.5
 = ±5
 = ±0.25 
 = ±0.5
 = ±1

We start out trying +1.25 using synthetic division:

1.25|4  1 -1    5 
    |   5  7.5  8.125 
     4  6  6.5 13.125

So 1.25 is not a solution

We try the next one -1.25

-1.25|4  1 -1  5 
     |  -5  5 -5 
      4 -4  4  0

Yes that's a solution because the remainder is 0.

So it factors as:

(x + 1.25)(4x² - 4x + 4) = 0

x + 1.25 = 0;       4x² - 4x + 4 = 0
       x = -1.25      x² - x + 1 = 0
                                
 
                               
                               
                               
                               

Solutions: -1.25 [or as a fraction ], , and .

Edwin