The sum of squares of all the real solutions of the equation _ (x+1) (2x^2+5x+3) = 4 - _ (2x+3) (x^2+2x+1) is equal to ________.

Step-by-step solution

For the logarithms to be defined, we must satisfy the following conditions: 1. Base of the first logarithm: x+1 > 0 x > -1 and x+1 1 x 0 . 2. Base of the second logarithm: 2x+3 > 0 x > - 3 2 and 2x+3 1 x -1 . 3. Arguments must be positive: 2x^2+5x+3 > 0 and x^2+2x+1 > 0 . Taking the intersection of all these conditions, the domain of the equation is x (-1, 0) (0, ) . Now, factorizing the arguments of the logarithms: 2x^2+5x+3 = (2x+3)(x+1) x^2+2x+1 = (x+1)^2 Substitute these into the given equation: _ (x+1) ((2x+3)