To solve this inequality, we will first break it down into four separate cases based on the different possible combinations of signs for x^2-5x+3 and x^2+2x-3.

Case 1: x^2-5x+3 > 0 and x^2+2x-3 > 0
In this case, both expressions inside the absolute value bars are positive. Therefore, the inequality simplifies to:
(x^2-5x+3) + (x^2+2x-3) < 10
2x^2 - 3 < 10
2x^2 < 13
x^2 < 6.5
-√6.5 < x < √6.5

Case 2: x^2-5x+3 > 0 and x^2+2x-3 < 0
In this case, the first expression is positive and the second one is negative. Therefore, the inequality simplifies to:
(x^2-5x+3) - (x^2+2x-3) < 10
-3x + 6 < 10
-3x < 4
x > -4/3

Case 3: x^2-5x+3 < 0 and x^2+2x-3 > 0
In this case, the first expression is negative and the second one is positive. Therefore, the inequality simplifies to:
-(x^2-5x+3) + (x^2+2x-3) < 10
3 < 10
This case is not valid because the inequality cannot be satisfied.

Case 4: x^2-5x+3 < 0 and x^2+2x-3 < 0
In this case, both expressions inside the absolute value bars are negative. Therefore, the inequality simplifies to:
-(x^2-5x+3) - (x^2+2x-3) < 10
-2x^2 + 8 < 10
-2x^2 < 2
x^2 > -1
This case is satisfied for all real values of x.

Therefore, the solution to the inequality |x^2-5x+3| + |x^2+2x-3| < 10 is -4/3 < x < √6.5.