To simplify the polynomial expression [tex]\(3x(-2x + 7) - 5(x - 1)(4x - 3)\)[/tex], let's go through the steps in detail:
1. Distribute in the first term [tex]\(3x(-2x + 7)\)[/tex]:
[tex]\[
3x \cdot (-2x) + 3x \cdot 7 = -6x^2 + 21x
\][/tex]
2. Expand and distribute in the second term [tex]\(-5(x - 1)(4x - 3)\)[/tex]:
To expand [tex]\((x - 1)(4x - 3)\)[/tex], apply the distributive property (FOIL method):
[tex]\[
(x - 1)(4x - 3) = x \cdot 4x + x \cdot (-3) - 1 \cdot 4x - 1 \cdot (-3)
\][/tex]
Simplifying each term gives:
[tex]\[
4x^2 - 3x - 4x + 3 = 4x^2 - 7x + 3
\][/tex]
Now, distribute the [tex]\(-5\)[/tex] across the expanded polynomial:
[tex]\[
-5 \cdot ( 4x^2 - 7x + 3) = -20x^2 + 35x - 15
\][/tex]
3. Combine the simplified results from steps 1 and 2:
[tex]\[
-6x^2 + 21x - 20x^2 + 35x - 15
\][/tex]
4. Combine like terms:
[tex]\[
(-6x^2 - 20x^2) + (21x + 35x) - 15
\][/tex]
[tex]\[
-26x^2 + 56x - 15
\][/tex]
This is our fully simplified expression. Now, we match it with the given options:
A. [tex]\(-26x^2 + 56x - 15\)[/tex]
B. [tex]\(14x^2 - 14x + 15\)[/tex]
C. [tex]\(-26x^2 + 21x - 15\)[/tex]
D. [tex]\(-2x^2 + 14x - 2\)[/tex]
The correct simplified polynomial expression [tex]\(-26x^2 + 56x - 15\)[/tex] matches option A.
Therefore, the correct answer is:
[tex]\[
A. -26x^2 + 56x - 15
\][/tex]
