$$ a ^ { 2 } - 2 a + a b - 2 b ^ { 2 } $$

Question

UpStudy AI · Accepted Answer

Let's simplify and attempt to factor the expression:

$$
a^{2} - 2a + ab - 2b^{2}
$$

**Step 1: Rearrange the Terms**

First, let's rearrange the terms to group similar terms together:

$$
a^{2} + ab - 2a - 2b^{2}
$$

**Step 2: Factor by Grouping**

Next, we can try to factor by grouping. Group the terms as follows:

$$
(a^{2} + ab) - (2a + 2b^{2})
$$

Now, factor out the greatest common factor (GCF) from each group:

$$
a(a + b) - 2(a + b^{2})
$$

At this point, we notice that the binomials inside the parentheses are not the same, and there isn't a common factor that can be factored out further. This indicates that the expression doesn't factor neatly with integer or simple rational coefficients.

**Step 3: Alternative Factoring Approach**

Alternatively, consider the expression as a quadratic in terms of $$ a $$:

$$
a^{2} + (b - 2)a - 2b^{2}
$$

To factor this quadratic, we look for two numbers that multiply to $$ -2b^{2} $$ (the constant term) and add up to $$ b - 2 $$ (the coefficient of the linear term).

However, no such pair of numbers exists that satisfies both conditions, which means the quadratic does not factor nicely over the integers.

**Conclusion**

Since the expression cannot be factored further using integer or simple rational coefficients, the simplified form of the expression is:

$$
a^{2} + ab - 2a - 2b^{2}
$$

\**Final Answer:***

The expression cannot be factored further and simplifies to:

$$
a^{2} + ab - 2a - 2b^{2}
$$
The expression cannot be factored further and simplifies to: $$ a^{2} + ab - 2a - 2b^{2} $$

\( a ^ { 2 } - 2 a + a b - 2 b ^ { 2 } \)

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