College

Simplify the following expression:

[tex]\left(5x^2 + 8x + 17\right)\left(5x^2 + 9x + 17\right)[/tex]

A. [tex]25x^4 + 85x^3 + 161x + 289[/tex]

B. [tex]25x^4 + 157x^2 + 289[/tex]

C. [tex]25x^4 + 85x^3 + 170x^2 + 136x + 289[/tex]

D. [tex]25x^4 + 85x^3 + 242x^2 + 289x + 289[/tex]

Answer :

To simplify the expression [tex]\((5x^2 + 8x + 17)(5x^2 + 9x + 17)\)[/tex], we can use the distributive property (also known as the FOIL method for binomials) to expand this product step-by-step.

### Step-by-Step Expansion:

1. Multiply the First Terms:
[tex]\[
5x^2 \times 5x^2 = 25x^4
\][/tex]

2. Multiply the Outer Terms:
[tex]\[
5x^2 \times 9x = 45x^3
\][/tex]

3. Multiply the Inner Terms:
[tex]\[
8x \times 5x^2 = 40x^3
\][/tex]

4. Multiply the Last Terms:
[tex]\[
8x \times 9x = 72x^2
\][/tex]

5. Multiply the First Inner Terms and First Last Terms:
[tex]\[
5x^2 \times 17 = 85x^2
\][/tex]

6. Multiply the Outer Last Terms:
[tex]\[
8x \times 17 = 136x
\][/tex]

7. Multiply the First and Second Constant Terms:
[tex]\[
17 \times 17 = 289
\][/tex]

### Combine Like Terms:

- The [tex]\(x^4\)[/tex] term: [tex]\(25x^4\)[/tex].
- The [tex]\(x^3\)[/tex] terms: [tex]\(45x^3 + 40x^3 = 85x^3\)[/tex].
- The [tex]\(x^2\)[/tex] terms: [tex]\(72x^2 + 85x^2 = 157x^2\)[/tex].
- The [tex]\(x\)[/tex] term: [tex]\(136x\)[/tex].
- The constant term: [tex]\(289\)[/tex].

### Final Simplified Expression:

Combining all these terms, we get:
[tex]\[
25x^4 + 85x^3 + 157x^2 + 136x + 289
\][/tex]

Thus, the correct answer is:
C. [tex]\(25x^4 + 85x^3 + 157x^2 + 136x + 289\)[/tex]

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