Question

Expand the following expressions:
1. (a - 2)(a squared +2a +4)
2. -(2x - 5)all to the power of 2
3. ( a over 2 + 1)( a over 2 - 1)
4. (1 - a to the power 4)(1 - a to the power of 2)(1 + a to the power of 2)

Answer 1

Sure! Let's expand each of the given expressions step by step.

### 1. \((a - 2)(a^2 + 2a + 4)\)

We will use the distributive property (also known as the FOIL method for two binomials):

\[
= a(a^2 + 2a + 4) - 2(a^2 + 2a + 4)
\]

Expanding both parts:

\[
= a^3 + 2a^2 + 4a - 2a^2 - 4a - 8
\]

Now, combine like terms:

\[
= a^3 + (2a^2 - 2a^2) + (4a - 4a) - 8 = a^3 - 8
\]

So, the expanded expression is:

\[
\boxed{a^3 - 8}
\]

### 2. \(- (2x - 5)^2\)

First, we calculate \((2x - 5)^2\):

\[
(2x - 5)(2x - 5) = 4x^2 - 10x - 10x + 25 = 4x^2 - 20x + 25
\]

Now, apply the negative:

\[
- (4x^2 - 20x + 25) = -4x^2 + 20x - 25
\]

Therefore, the expanded expression is:

\[
\boxed{-4x^2 + 20x - 25}
\]

### 3. \(\left(\frac{a}{2} + 1\right)\left(\frac{a}{2} - 1\right)\)

This expression follows the difference of squares formula:

\[
= \left(\frac{a}{2}\right)^2 - (1)^2
\]

Calculating each part:

\[
= \frac{a^2}{4} - 1
\]

Thus, the expanded expression is:

\[
\boxed{\frac{a^2}{4} - 1}
\]

### 4. \((1 - a^4)(1 - a^2)(1 + a^2)\)

First, expand \((1 - a^2)(1 + a^2)\) using the difference of squares:

\[
= 1^2 - (a^2)^2 = 1 - a^4
\]

Now substitute that back in:

\[
(1 - a^4)(1 - a^4)
\]

Now expand this:

\[
= (1 - a^4)^2 = 1^2 - 2(1)(a^4) + (a^4)^2 = 1 - 2a^4 + a^8
\]

Finally, the expanded expression is:

\[
\boxed{a^8 - 2a^4 + 1}
\]

All four expressions have been expanded successfully!