Simplify fully
\[
\frac{6 x^{2}+15 x}{12 x^{2}-75}
\]

Step 1 ：Find the greatest common divisor (GCD) of the coefficients in the numerator and the denominator.

Step 2 ：\(\text{numerator coefficients} = [6, 15]\)

Step 3 ：\(\text{denominator coefficients} = [12, -75]\)

Step 4 ：\(\text{numerator GCD} = 3\)

Step 5 ：\(\text{denominator GCD} = 3\)

Step 6 ：Divide both the numerator and the denominator by their respective GCDs to simplify the expression.

Step 7 ：\(\text{simplified numerator} = [2, 5]\)

Step 8 ：\(\text{simplified denominator} = [4, -25]\)

Step 9 ：Rewrite the expression with the simplified coefficients and check if there are any common factors between the numerator and the denominator that can be further simplified.

Step 10 ：\(\frac{2x^2 + 5x}{4x^2 - 25} = \frac{x(2x + 5)}{x(4x - 25)}\)

Step 11 ：Cancel out the common factor \(x\) from the numerator and the denominator.

Step 12 ：\(\frac{x(2x + 5)}{x(4x - 25)} = \frac{2x + 5}{4x - 25}\)

Step 13 ：Final Answer: \(\boxed{\frac{2x + 5}{4x - 25}}\)