3c2+4c-4/9c2-4

To simplify the expression \( \frac{3c^2 + 4c - 4}{9c^2 - 4} \), we can factor both the numerator and the denominator if possible.

1. Factor the numerator \( 3c^2 + 4c - 4 \).

To factor, we can look for two numbers that multiply to \( 3 \times (-4) = -12 \) and add to \( 4 \). The numbers \( 6 \) and \( -2 \) work.

Thus, we rewrite the expression: \[ 3c^2 + 6c - 2c - 4 \] Next, we group the terms: \[ (3c^2 + 6c) + (-2c - 4) \] Factor by grouping: \[ 3c(c + 2) - 2(c + 2) \] Now we can factor out the common factor \( (c + 2) \): \[ (3c - 2)(c + 2) \]

2. Factor the denominator \( 9c^2 - 4 \).

This is a difference of squares: \[ 9c^2 - 4 = (3c)^2 - 2^2 = (3c - 2)(3c + 2) \]

3. Combine the results: The expression becomes: \[ \frac{(3c - 2)(c + 2)}{(3c - 2)(3c + 2)} \]

4. Cancel the common factors: The \( (3c - 2) \) terms in the numerator and denominator cancel out, giving us: \[ \frac{c + 2}{3c + 2} \]

So, the simplified form of the original expression \( \frac{3c^2 + 4c - 4}{9c^2 - 4} \) is: \[ \frac{c + 2}{3c + 2} \] (as long as \( 3c - 2 \neq 0 \), which means \( c \neq \frac{2}{3} \)).