(2r+3)(2r-1)=-(3r+1)

Solution for (2r+3)(2r-1)=-(3r+1) equation:

Simplifying
(2r + 3)(2r + -1) = -1(3r + 1)

Reorder the terms:
(3 + 2r)(2r + -1) = -1(3r + 1)

Reorder the terms:
(3 + 2r)(-1 + 2r) = -1(3r + 1)

Multiply (3 + 2r) * (-1 + 2r)
(3(-1 + 2r) + 2r * (-1 + 2r)) = -1(3r + 1)
((-1 * 3 + 2r * 3) + 2r * (-1 + 2r)) = -1(3r + 1)
((-3 + 6r) + 2r * (-1 + 2r)) = -1(3r + 1)
(-3 + 6r + (-1 * 2r + 2r * 2r)) = -1(3r + 1)
(-3 + 6r + (-2r + 4r2)) = -1(3r + 1)

Combine like terms: 6r + -2r = 4r
(-3 + 4r + 4r2) = -1(3r + 1)

Reorder the terms:
-3 + 4r + 4r2 = -1(1 + 3r)
-3 + 4r + 4r2 = (1 * -1 + 3r * -1)
-3 + 4r + 4r2 = (-1 + -3r)

Solving
-3 + 4r + 4r2 = -1 + -3r

Solving for variable 'r'.

Reorder the terms:
-3 + 1 + 4r + 3r + 4r2 = -1 + -3r + 1 + 3r

Combine like terms: -3 + 1 = -2
-2 + 4r + 3r + 4r2 = -1 + -3r + 1 + 3r

Combine like terms: 4r + 3r = 7r
-2 + 7r + 4r2 = -1 + -3r + 1 + 3r

Reorder the terms:
-2 + 7r + 4r2 = -1 + 1 + -3r + 3r

Combine like terms: -1 + 1 = 0
-2 + 7r + 4r2 = 0 + -3r + 3r
-2 + 7r + 4r2 = -3r + 3r

Combine like terms: -3r + 3r = 0
-2 + 7r + 4r2 = 0

Factor a trinomial.
(-2 + -1r)(1 + -4r) = 0

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

Simplifying
-2 + -1r = 0

Solving
-2 + -1r = 0

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

Add '2' to each side of the equation.
-2 + 2 + -1r = 0 + 2

Combine like terms: -2 + 2 = 0
0 + -1r = 0 + 2
-1r = 0 + 2

Combine like terms: 0 + 2 = 2
-1r = 2

Divide each side by '-1'.
r = -2

Simplifying
r = -2
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 = {-2, 0.25}