Okay, let me try to figure out how to solve this equation. The equation given is (6r² - 19r -7)/(2r -7) = 4r -3. Hmm, so I need to find the value of r that satisfies this equation. 

First, maybe I should start by simplifying the left side of the equation. Since there's a denominator, 2r -7, perhaps I can perform polynomial division or factor the numerator to see if it simplifies with the denominator. Let me check if the numerator factors. The numerator is 6r² -19r -7. Let's try factoring that.

To factor 6r² -19r -7, I need two numbers that multiply to (6)(-7) = -42 and add up to -19. Let's see, factors of -42...  -21 and 2? Because -21 + 2 = -19. Yeah, that works. So, I can rewrite the middle term using these numbers.

So, 6r² -21r +2r -7. Now, group the terms: (6r² -21r) + (2r -7). Factor out the common factors from each group: 3r(2r -7) +1(2r -7). Now, factor out (2r -7): (3r +1)(2r -7). 

Oh, that's nice! So the numerator factors to (3r +1)(2r -7). Then the original fraction becomes (3r +1)(2r -7)/(2r -7). Since the denominator is 2r -7, as long as 2r -7 ≠ 0, we can cancel out the (2r -7) terms. So, this simplifies to 3r +1, provided that 2r -7 ≠ 0, which means r ≠ 7/2.

So now, the equation simplifies to 3r +1 = 4r -3. Let's solve this equation. Subtract 3r from both sides: 1 = r -3. Then add 3 to both sides: r = 4. 

But wait, I need to check if r =4 makes the original denominator zero. The denominator was 2r -7. Plugging r=4: 2*4 -7 =8 -7=1≠0. So, r=4 is a valid solution. 

But hold on, maybe I should verify the solution by plugging it back into the original equation to make sure I didn't make any mistakes during the simplification.

Original equation: (6r² -19r -7)/(2r -7) = 4r -3. Let's plug r=4 into the left side:

Numerator: 6*(4)^2 -19*4 -7 =6*16 -76 -7=96 -76 -7=13.

Denominator:2*4 -7=8-7=1. So left side is 13/1=13.

Right side:4*4 -3=16-3=13. Both sides equal 13, so it checks out.

Therefore, r=4 is indeed the solution.

Wait, but just to make sure there are no other solutions. Since we canceled out the (2r -7) term, which is valid only when 2r -7 ≠0, but if 2r -7=0, then the original equation becomes undefined. So, there's no solution when 2r -7=0, which is r=7/2. But plugging r=7/2 into the original equation: numerator is 6*(7/2)^2 -19*(7/2) -7. Let's compute that:

6*(49/4) - (133/2) -7 = (294/4) - (133/2) -7 = (147/2) - (133/2) -7 = (14/2) -7 =7 -7=0. So numerator is 0, denominator is 0. So it's 0/0, which is undefined. Therefore, r=7/2 is not a valid solution. Hence, the only solution is r=4.

So, after factoring, simplifying, solving, and verifying, the answer is r=4.
Therefore, the value of $r$ that satisfies the equation is $\boxed{4}$.
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