3x²+7x-6/ 4-9x²

To simplify the expression \(\frac{3x^2 + 7x - 6}{4 - 9x^2}\), we'll first factor both the numerator and the denominator, if possible. **Step 1: Factor the numerator \(3x^2 + 7x - 6\)** To factor this quadratic expression, we will look for two numbers that multiply to \(3 \times -6 = -18\) and add to \(7\). The two numbers that satisfy these conditions are \(9\) and \(-2\). Now, we can rewrite the middle term: \[ 3x^2 + 9x - 2x - 6 \] Now, we can factor by grouping: \[ (3x^2 + 9x) + (-2x - 6) \] This gives: \[ 3x(x + 3) - 2(x + 3) \] Now, factor out \((x + 3)\): \[ (3x - 2)(x + 3) \] **Step 2: Factor the denominator \(4 - 9x^2\)** This expression is a difference of squares and can be factored as follows: \[ 4 - 9x^2 = (2)^2 - (3x)^2 = (2 - 3x)(2 + 3x) \] **Step 3: Write the fully factored form of the expression** Now we can substitute the factored forms back into the expression: \[ \frac{3x^2 + 7x - 6}{4 - 9x^2} = \frac{(3x - 2)(x + 3)}{(2 - 3x)(2 + 3x)} \] **Step 4: Simplify** There are no common factors to cancel out in \((3x - 2)(x + 3)\) and \((2 - 3x)(2 + 3x)\). Note that \(2 - 3x\) can be rewritten as \(-(3x - 2)\), allowing us to simplify: \[ \frac{(3x - 2)(x + 3)}{-(3x - 2)(2 + 3x)} = -\frac{x + 3}{2 + 3x} \] **Final Result:** Thus, the simplified form of the expression is \[ -\frac{x + 3}{2 + 3x} \] or you can also write it as \[ -\frac{x + 3}{3x + 2} \] depending on the order.