To simplify the given expression \((-5x^2+7)(-5x^2-7) - 4 (x^3+1)(2x+3)\), let's break it down step by step.
Step 1: Expand \((-5x^2+7)(-5x^2-7)\)
Using the difference of squares formula \((a-b)(a+b) = a^2 - b^2\):
\[ (-5x^2 + 7)(-5x^2 - 7) = (5x^2)^2 - 7^2 = 25x^4 - 49 \]
Step 2: Expand \(4(x^3 + 1)(2x + 3)\)
First calculate \((x^3 + 1)(2x + 3)\):
Using the distributive property:
\[ (x^3 + 1)(2x + 3) = x^3(2x + 3) + 1(2x + 3) = 2x^4 + 3x^3 + 2x + 3 \]
Now multiply by 4:
\[ 4(2x^4 + 3x^3 + 2x + 3) = 8x^4 + 12x^3 + 8x + 12 \]
Step 3: Combine the results
Now, substitute back the expanded expressions into the original equation:
\[ 25x^4 - 49 - (8x^4 + 12x^3 + 8x + 12) \]
Distributing the negative sign:
\[ 25x^4 - 49 - 8x^4 - 12x^3 - 8x - 12 \]
Now combine the like terms:
-
Combine the \(x^4\) terms: \[ 25x^4 - 8x^4 = 17x^4 \]
-
Combine the \(x^3\) terms: \[ -12x^3 \]
-
Combine the \(x\) terms: \[ -8x \]
-
Combine the constant terms: \[ -49 - 12 = -61 \]
Putting it all together gives:
\[ 17x^4 - 12x^3 - 8x - 61 \]
Final Answer:
The simplified expression is \(17x^4 - 12x^3 - 8x - 61\).
The correct choice from the options provided is:
C: \(17x^4 - 12x^3 - 8x - 61\).