To solve the equation [tex]\((x-2)\left(x^2+2x+4\right) = x^3 - 4\)[/tex], let's expand and simplify both sides of the equation to see if they are equal.
First, we will expand the left side of the equation:
Given the expression on the left side: [tex]\((x-2)(x^2 + 2x + 4)\)[/tex], we will use the distributive property to expand this:
- Multiply [tex]\(x\)[/tex] by each term in [tex]\((x^2 + 2x + 4)\)[/tex]:
[tex]\[
x \cdot (x^2 + 2x + 4) = x^3 + 2x^2 + 4x
\][/tex]
- Multiply [tex]\(-2\)[/tex] by each term in [tex]\((x^2 + 2x + 4)\)[/tex]:
[tex]\[
-2 \cdot (x^2 + 2x + 4) = -2x^2 - 4x - 8
\][/tex]
Now, combine the results from both multiplications:
[tex]\[
(x-2)(x^2 + 2x + 4) = x^3 + 2x^2 + 4x - 2x^2 - 4x - 8
\][/tex]
Next, we simplify the expanded terms by combining like terms:
[tex]\[
x^3 + 2x^2 + 4x - 2x^2 - 4x - 8 = x^3 - 8
\][/tex]
So, the expanded form of the left side is [tex]\(x^3 - 8\)[/tex].
Next, we will compare this with the right side of the equation:
[tex]\[
x^3 - 4
\][/tex]
We can see that the expanded left side [tex]\(x^3 - 8\)[/tex] is not equal to the right side [tex]\(x^3 - 4\)[/tex].
Thus, the equation [tex]\((x-2)(x^2+2x+4) = x^3 - 4\)[/tex] does not hold true.
The results confirm that the left side [tex]\(x^3 - 8\)[/tex] is not equal to the right side [tex]\(x^3 - 4\)[/tex], hence the equation is false.
