Uiz: Copy of Unit 7 Lesson 5 12/6/24 Simplify. Express your answer as a single fraction in simplest form. \( x+\frac{9}{4 x^{2}+20 x-12} \)

To simplify the expression \( x+\frac{9}{4x^{2}+20x-12} \), we first need to factor the quadratic in the denominator. The expression in the denominator can be factored as \( 4(x^2 + 5x - 3) \). Next, we rewrite \( x \) with a common denominator: \[ x = \frac{x(4x^2 + 20x - 12)}{4x^2 + 20x - 12} = \frac{4x^3 + 20x^2 - 12x}{4x^2 + 20x - 12} \] Combine the fractions: \[ \frac{4x^3 + 20x^2 - 12x + 9}{4x^2 + 20x - 12} \] Now, let's simplify the numerator \( 4x^3 + 20x^2 - 12x + 9 \) if possible. This can be a bit complex, but checking for factorization or common factors might help simplify further. After potential simplifications, make sure to present the final result as one fraction in simplest form. Now, the complete expression would look like: \[ \frac{4x^3 + 20x^2 - 12x + 9}{4x^2 + 20x - 12} \] And you can check if the numerator can be factored or simplified further depending on specific values of \( x \).