Verify the Equation: x⁴-16 = (x²-4)(4+x²) - Polynomial Identity Check

To solve this problem, let's follow these steps:

• Step 1: Factor the left-hand side $x^4 - 16$ using the difference of squares.
• Step 2: Expand the right-hand side expression.
• Step 3: Verify if both sides are equivalent after simplification.

Now, let's evaluate each step:

Step 1:
The expression on the left-hand side is $x^4 - 16$.
This can be viewed as $(x^2)^2 - 4^2$,
which is a difference of squares. Therefore, we can factor it as:
$(x^2 - 4)(x^2 + 4)$.

Step 2:
Next, consider the right-hand side, $(x^2 - 4)(4 + x^2)$.
To expand, use distribution (FOIL method):
- First: $x^2 \times 4 = 4x^2$
- Outer: $x^2 \times x^2 = x^4$
- Inner: $-4 \times 4 = -16$
- Last: $-4 \times x^2 = -4x^2$
Combine these terms:
$x^4 + 4x^2 - 16 - 4x^2 = x^4 - 16$.

Step 3:
The expanded term, $x^4 - 16$, matches the factored left-hand side expression, $(x^2-4)(x^2+4)$, showing that both sides are equivalent.

Therefore, the equation $x^4 - 16 = (x^2 - 4)(4 + x^2)$ is True for all values of $x$.