Find the simplified form of `(-10x^2- 35x + 20)/(3x^2+12x)`.

Options

A. `(-5(2x-1))/(3x)`
B. `(-5)/3`
C. `(-5(2x^2+7x+4))/(3x^2+ 12x)`
D. `-7x^2-23x+20`

Answer

To simplify the expression \((-10x^2 - 35x + 20)/(3x^2 + 12x)\), we can follow these steps:

### Step 1: Factor the numerator and the denominator
First, we need to factor both the numerator and the denominator.

**Numerator:**
\[ -10x^2 - 35x + 20 \]
We can rewrite \(-35x\) as \(-40x + 5x\):
\[ -10x^2 - 40x + 5x + 20 \]

Now, we can group the terms:
\[ (-10x^2 - 40x) + (5x + 20) \]

Factoring out common factors from each group:
\[ -10x(x + 4) + 5(x + 4) \]

Now, we can factor out \((x + 4)\):
\[ (x + 4)(-10x + 5) \]

**Denominator:**
\[ 3x^2 + 12x \]
We can factor out \(3x\):
\[ 3x(x + 4) \]

### Step 2: Rewrite the expression
Now we can rewrite the original expression using the factored forms:
\[ \frac{(x + 4)(-10x + 5)}{3x(x + 4)} \]

### Step 3: Cancel common factors
The \((x + 4)\) in the numerator and denominator can be canceled:
\[ \frac{-10x + 5}{3x} \]

### Step 4: Simplify the expression
Now we can simplify the numerator:
\[ -10x + 5 = -5(2x - 1) \]

Thus, the expression becomes:
\[ \frac{-5(2x - 1)}{3x} \]

### Final Result
The simplified form of the given expression is:
\[ \frac{-5(2x - 1)}{3x} \]