Factor polynomal $$ b^{2}-2 b^{2}+3 b-6 $$

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To factor the polynomial $$ b^{2} - 2b^{2} + 3b - 6 $$, let's follow these steps:

### 1. **Combine Like Terms**

First, simplify the expression by combining like terms:

$$
b^{2} - 2b^{2} + 3b - 6 = (-b^{2}) + 3b - 6
$$

So, the expression simplifies to:

$$
-b^{2} + 3b - 6
$$

### 2. **Factor Out the Common Factor**

Notice that each term in the simplified expression has a common factor of $$-1$$. Factoring this out gives:

$$
- (b^{2} - 3b + 6)
$$

### 3. **Attempt to Factor the Quadratic**

Now, let's attempt to factor the quadratic expression inside the parentheses:

$$
b^{2} - 3b + 6
$$

To factor this, we look for two numbers that multiply to $$6$$ (the constant term) and add up to $$-3$$ (the coefficient of the middle term). However, such real numbers do not exist because the discriminant of this quadratic equation is negative:

$$
\text{Discriminant} = (-3)^2 - 4 \times 1 \times 6 = 9 - 24 = -15
$$

A negative discriminant indicates that the quadratic cannot be factored into real numbers.

### 4. **Final Factored Form**

Since the quadratic doesn't factor nicely over the real numbers, the fully factored form of the original polynomial is:

$$
- (b^{2} - 3b + 6)
$$

### **Conclusion**

The polynomial $$ b^{2} - 2b^{2} + 3b - 6 $$ simplifies and factors as:

$$
- (b^{2} - 3b + 6)
$$

This is the most factored form using real numbers.
The polynomial $$ b^{2} - 2b^{2} + 3b - 6 $$ simplifies to $$ - (b^{2} - 3b + 6) $$.

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