Let's simplify each expression step by step.
### Expression 6:
[tex]\[
(2 + a^2 - 2a - a^3)(a + 1)
\][/tex]
First, we distribute each term inside the first parenthesis by the term [tex]\(a + 1\)[/tex]:
[tex]\[
= (2 + a^2 - 2a - a^3) \times a + (2 + a^2 - 2a - a^3) \times 1
\][/tex]
Distributing [tex]\(a\)[/tex] to each term inside the first parenthesis:
[tex]\[
= 2a + a^3 - 2a^2 - a^4
\][/tex]
Next, distributing 1 to each term inside the parenthesis:
[tex]\[
= 2 + a^2 - 2a - a^3
\][/tex]
Now, add these two results together:
[tex]\[
2a + a^3 - 2a^2 - a^4 + 2 + a^2 - 2a - a^3
\][/tex]
Combine like terms:
[tex]\[
= -a^4 + (a^3 - a^3) + (-2a^2 + a^2) + (2a - 2a) + 2
\][/tex]
[tex]\[
= -a^4 - a^2 + 2
\][/tex]
So, the simplified form is:
[tex]\[
(2 + a^2 - 2a - a^3)(a + 1) = -a^4 - a^2 + 2
\][/tex]
### Expression 7:
[tex]\[
\left(\frac{1}{2} x^2 - 2x + \frac{1}{4}\right)\left(-\frac{1}{2}\right)
\][/tex]
Distribute [tex]\( -\frac{1}{2} \)[/tex] to each term inside the parenthesis:
[tex]\[
= \left(\frac{1}{2} x^2 \times -\frac{1}{2}\right) + \left(-2x \times -\frac{1}{2}\right) + \left(\frac{1}{4} \times -\frac{1}{2}\right)
\][/tex]
Simplify each term:
[tex]\[
= -\frac{1}{4} x^2 + x - \frac{1}{8}
\][/tex]
So, the simplified form is:
[tex]\[
\left(\frac{1}{2} x^2 - 2x + \frac{1}{4}\right)\left(-\frac{1}{2}\right) = -\frac{1}{4} x^2 + x - \frac{1}{8}
\][/tex]
Therefore, the final answers are:
6. [tex]\(\left(2 + a^2 - 2a - a^3\right)(a+1) = -a^4 - a^2 + 2\)[/tex]
7. [tex]\(\left(\frac{1}{2} x^2 - 2 x + \frac{1}{4}\right)\left(- \frac{1}{2}\right) = -\frac{1}{4} x^2 + x - \frac{1}{8}\)[/tex]
