(4r-1)(r+1)=0

Solution for (4r-1)(r+1)=0 equation:

Simplifying
(4r + -1)(r + 1) = 0

Reorder the terms:
(-1 + 4r)(r + 1) = 0

Reorder the terms:
(-1 + 4r)(1 + r) = 0

Multiply (-1 + 4r) * (1 + r)
(-1(1 + r) + 4r * (1 + r)) = 0
((1 * -1 + r * -1) + 4r * (1 + r)) = 0
((-1 + -1r) + 4r * (1 + r)) = 0
(-1 + -1r + (1 * 4r + r * 4r)) = 0
(-1 + -1r + (4r + 4r2)) = 0

Combine like terms: -1r + 4r = 3r
(-1 + 3r + 4r2) = 0

Solving
-1 + 3r + 4r2 = 0

Solving for variable 'r'.

Factor a trinomial.
(-1 + -1r)(1 + -4r) = 0

Subproblem 1
Set the factor '(-1 + -1r)' equal to zero and attempt to solve:

Simplifying
-1 + -1r = 0

Solving
-1 + -1r = 0

Move all terms containing r to the left, all other terms to the right.

Add '1' to each side of the equation.
-1 + 1 + -1r = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -1r = 0 + 1
-1r = 0 + 1

Combine like terms: 0 + 1 = 1
-1r = 1

Divide each side by '-1'.
r = -1

Simplifying
r = -1
Subproblem 2
Set the factor '(1 + -4r)' equal to zero and attempt to solve:

Simplifying
1 + -4r = 0

Solving
1 + -4r = 0

Move all terms containing r to the left, all other terms to the right.

Add '-1' to each side of the equation.
1 + -1 + -4r = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -4r = 0 + -1
-4r = 0 + -1

Combine like terms: 0 + -1 = -1
-4r = -1

Divide each side by '-4'.
r = 0.25

Simplifying
r = 0.25
Solution
r = {-1, 0.25}