Simplify (9a² - 4)/(2 - 3a) - (6a² - 5a - 6)/(3 - 2a).

Step-by-step solution

1. 1

   Plan the route: factor everything before finding a common denominator. Both denominators are written 'backwards' relative to the factors that will appear on top, and that reversal is worth exactly a factor of $-1$. Spotting it first avoids a messy common-denominator computation.

2. 2

   Factor and reduce the first fraction. $9a^2-4 = (3a)^2-2^2 = (3a-2)(3a+2)$, while $2-3a = -(3a-2)$. Cancelling the matching factor: $\frac{(3a-2)(3a+2)}{-(3a-2)} = -(3a+2) = -3a-2, \qquad a \neq \tfrac23.$

3. 3

   Factor the second numerator. For $6a^2-5a-6$ find integers with product $6(-6) = -36$ and sum $-5$: those are $4$ and $-9$. Splitting and grouping: $6a^2+4a-9a-6 = 2a(3a+2)-3(3a+2) = (3a+2)(2a-3)$.

4. 4

   Reduce the second fraction. Since $3-2a = -(2a-3)$, $\frac{(3a+2)(2a-3)}{-(2a-3)} = -(3a+2) = -3a-2, \qquad a \neq \tfrac32.$ Both fractions have reduced to the same expression.

5. 5

   Subtract. $(-3a-2)-(-3a-2) = -3a-2+3a+2 = 0.$ The expression is identically zero wherever it is defined.

6. 6

   Record the excluded values and spot-check. The original expression is undefined at $a=\tfrac23$ and $a=\tfrac32$. Substituting $a=0.3$, $1.7$, $-2$ and $5$ all give $0$ to full precision, confirming the identity.