To simplify the expression \( \frac{x^2 - 9x^2 - 4 \cdot 4x - 8}{12x + 36} \), we first simplify the numerator.
Starting with \( x^2 - 9x^2 \): \[ x^2 - 9x^2 = -8x^2 \]
Next, we simplify \( -4 \cdot 4x - 8 \): \[ -4 \cdot 4x = -16x \] Thus, the numerator combines as: \[ -8x^2 - 16x - 8 \]
So, we now rewrite the expression as: \[ \frac{-8x^2 - 16x - 8}{12x + 36} \]
Next, we can factor out the common factor in the numerator and the denominator. First, we can factor out \(-8\) from the numerator: \[ -8(x^2 + 2x + 1) = -8(x + 1)^2 \]
Now, let’s look at the denominator \(12x + 36\): \[ 12x + 36 = 12(x + 3) \]
Putting it all together, we have: \[ \frac{-8(x + 1)^2}{12(x + 3)} \]
We can simplify this fraction by dividing both the numerator and the denominator by 4: \[ \frac{-2(x + 1)^2}{3(x + 3)} \]
So, the polynomial expression in simplest form is: \[ \frac{-2(x + 1)^2}{3(x + 3)} \]