Let's simplify the expression [tex]\(-2a \cdot (2a - 5bc + 4b^2)\)[/tex] step-by-step.
### Step 1: Distribute the [tex]\(-2a\)[/tex] across each term inside the parentheses.
The expression inside the parentheses is [tex]\(2a - 5bc + 4b^2\)[/tex]. We need to multiply [tex]\(-2a\)[/tex] by each term in the parentheses separately:
- Multiply [tex]\(-2a\)[/tex] by [tex]\(2a\)[/tex]:
[tex]\[
-2a \cdot 2a = -4a^2
\][/tex]
- Multiply [tex]\(-2a\)[/tex] by [tex]\(-5bc\)[/tex]:
[tex]\[
-2a \cdot (-5bc) = 10abc
\][/tex]
- Multiply [tex]\(-2a\)[/tex] by [tex]\(4b^2\)[/tex]:
[tex]\[
-2a \cdot 4b^2 = -8ab^2
\][/tex]
### Step 2: Combine all these terms.
Now, we'll combine the results from the distribution:
- [tex]\(-4a^2\)[/tex]
- [tex]\(10abc\)[/tex]
- [tex]\(-8ab^2\)[/tex]
Putting them all together, we get:
[tex]\[
-4a^2 + 10abc - 8ab^2
\][/tex]
### Step 3: Group the expression to show the common factor.
While this is indeed a simplified form, we notice the potential to factor it differently. Observing the common terms, [tex]\(-4a^2\)[/tex], [tex]\(10abc\)[/tex], and [tex]\(-8ab^2\)[/tex], let's try to identify a common factor approach.
We can rearrange and factor differently:
[tex]\[
= 2a (2a - 4b^2 + 5bc)
\][/tex]
This represents the simplified form of the original expression, reflecting combining and factoring principles.
Thus, the simplified version of the original expression [tex]\(-2a \cdot (2a - 5bc + 4b^2)\)[/tex] is:
[tex]\[
2a(-2a - 4b^2 + 5bc)
\][/tex]
This concludes our detailed step-by-step simplification process.
