Write down the equations of the three asymptotes and the coordinates of the points where the curve y = (3x+2)(x-3)/(x-2)(x+1) crosses the axes.

Step 1: find the asymptotesThere are two kinds of asymptotes we look for: vertical and horizontal.The vertical ones happen when y goes to negative or positive infinity, which is when the denominator of the fraction is 0. Here, the denominator of the fraction is 0 when x = 2, and x = -1.The horizontal ones happen when x goes to negative or positive infinity. We can find out the value of y by looking at what the fraction tends to when x goes to infinity or negative infinity. We can make arguments like this: 3x+2 is approximately equal to 3x when x is large, and x-3, x-2, x+1 are approximately equal to x for the same reason. So we can put that into the fraction, and say that y is approximately equal to 3x * x / x * x = 3 when x is negative/positive infinity. So the horizontal asymptote is at y = 3.Overall answer to step 1: x = 2, x = -1, y = 3Step 2: Coordinates of the points where the curve touches the axisThe curve touches the axis when x=0 or y=0.When x=0, y = (2)(-3)/(-2)(+1) = -6 / -2 = 3. So the coordinate is (0,3).When y=0, 0 = (3x+2)(x-3)/(x-2)(x+1). Multiplying both sides by (x-2)(x+1), we seethat 0 = (3x+2)(x-3). We can expand the brackets to find that this is just a quadratic equation: 0 = 3x^2 - 7x - 6. Solving this, we get x = 3 and x = -2/3. So the coordinates are (3, 0), (-2/3, 0). Overall answer to step 2: (0,3), (3,0), (-2/3, 0)