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-2^4 - (x^2 - 1)^2 =...

`-2^4 - (x^2 - 1)^2` =

A

`-x^4 + 2x^2 + 15`

B

`-x^4 - 2x^2 + 17`

C

`-x^4 + 2x^2 - 17`

D

`-x^4 + 2x^2 - 15`

Text Solution

AI Generated Solution

The correct Answer is:
To solve the expression \(-2^4 - (x^2 - 1)^2\), we will break it down step by step. ### Step 1: Evaluate \(-2^4\) First, we calculate \(2^4\): \[ 2^4 = 16 \] Thus, \(-2^4 = -16\). ### Step 2: Rewrite the expression Now we can rewrite the expression: \[ -2^4 - (x^2 - 1)^2 = -16 - (x^2 - 1)^2 \] ### Step 3: Expand \((x^2 - 1)^2\) Next, we will expand \((x^2 - 1)^2\) using the formula \((a - b)^2 = a^2 - 2ab + b^2\): \[ (x^2 - 1)^2 = (x^2)^2 - 2(x^2)(1) + (1)^2 = x^4 - 2x^2 + 1 \] ### Step 4: Substitute the expansion back into the expression Now we substitute the expanded form back into the expression: \[ -16 - (x^4 - 2x^2 + 1) \] ### Step 5: Distribute the negative sign Distributing the negative sign gives us: \[ -16 - x^4 + 2x^2 - 1 \] ### Step 6: Combine like terms Now, we combine the constant terms: \[ -16 - 1 = -17 \] So the expression simplifies to: \[ - x^4 + 2x^2 - 17 \] ### Final Answer Thus, the final simplified expression is: \[ - x^4 + 2x^2 - 17 \] ---
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