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.