Question

Perform the operation(s) and simplify completely.
(-5x^2+7)(-5x^2-7) -4 (x^3+1)(2x+3)









A
17x4+12x3+8x−3717x^4+12x^3+8x-3717x
4
+12x
3
+8x−37

B
17x4+3x2−2x−5217x^4+3x^2-2x-5217x
4
+3x
2
−2x−52

C
17x4−12x3−8x−6117x^4-12x^3-8x-6117x
4
−12x
3
−8x−61

D
−33x4−12x3−8x+37-33x^4-12x^3-8x+37−33x
4
−12x
3
−8x+37

Answer 1

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:

1. Combine the \(x^4\) terms:
\[
25x^4 - 8x^4 = 17x^4
\]

2. Combine the \(x^3\) terms:
\[
-12x^3
\]

3. Combine the \(x\) terms:
\[
-8x
\]

4. 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: \(17x^4 - 12x^3 - 8x - 61\)**.