Explanation:
To find the roots of the quadratic equation 2x^2 + 3x - 9 = 0, we can use the quadratic formula:

x = (-b ± sqrt(b^2 - 4ac)) / 2a

where a, b, and c are the coefficients of the quadratic equation.

Step 1:
Identify the values of a, b, and c from the given quadratic equation.
a = 2, b = 3, c = -9

Step 2:
Substitute the values of a, b, and c in the quadratic formula.

x = (-3 ± sqrt(3^2 - 4(2)(-9))) / 2(2)

Simplifying the equation, we get:

x = (-3 ± sqrt(105)) / 4

Step 3:
Now we need to simplify the square root of 105.

105 = 3 x 5 x 7

We can simplify the square root of 105 as:

sqrt(105) = sqrt(3 x 5 x 7) = sqrt(3) x sqrt(5) x sqrt(7)

Step 4:
Substitute the simplified value of square root of 105 in the quadratic formula.

x = (-3 ± sqrt(3) x sqrt(5) x sqrt(7)) / 4

Now we can simplify further by dividing the numerator and denominator by 2.

x = (-3/2) ± (sqrt(3) x sqrt(35)) / 4

Step 5:
We can simplify the expression by separating it into two roots.

x = (-3/2) + (sqrt(3) x sqrt(35)) / 4 or x = (-3/2) - (sqrt(3) x sqrt(35)) / 4

Step 6:
We can further simplify the expression by dividing the numerator and denominator of each root by 2.

x = (-3/4) + (sqrt(3) x sqrt(35)) / 8 or x = (-3/4) - (sqrt(3) x sqrt(35)) / 8

Step 7:
Simplify the expression by finding the common denominator.

x = (-6 + sqrt(3 x 35)) / 8 or x = (-6 - sqrt(3 x 35)) / 8

Step 8:
Simplify further by multiplying and dividing the numerator of each root by 2.

x = (-6 + sqrt(105)) / 8 or x = (-6 - sqrt(105)) / 8

Step 9:
Now we can see that the roots are 1.5 and -3.

x = (1.5) or x = (-3)

Therefore, the correct answer is option B, which is 1.5 and -3.