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------------------------------------------------ Factor [tex]$x^4 + 17x^2 + 70$[/tex] completely.

All factors in your answer should have integer coefficients.

Certainly! Let's factor the polynomial [tex]$x^4 + 17x^2 + 70$[/tex] completely.

### Step-by-Step Solution

1. Understanding the Polynomial:
- The given polynomial is [tex]$x^4 + 17x^2 + 70$[/tex].
- Notice that this can be expressed as a quadratic in terms of [tex]$x^2$[/tex]. Let [tex]$y = x^2$[/tex]. Substituting for [tex]$y$[/tex], we have:
[tex]\[
y^2 + 17y + 70
\][/tex]
- Now, the polynomial looks like a standard quadratic equation in [tex]$y$[/tex].

2. Factoring the Quadratic:
- We need two numbers that multiply to 70 and add up to 17.
- The numbers 7 and 10 satisfy [tex]$ (7 \times 10 = 70) $[/tex] and [tex]$ (7 + 10 = 17) $[/tex].

3. Writing the Factors:
- Using the numbers found above, we can write the quadratic in its factored form as:
[tex]\[
(y + 7)(y + 10)
\][/tex]
- Replace [tex]$y$[/tex] with [tex]$x^2$[/tex] to return to the original variable:
[tex]\[
(x^2 + 7)(x^2 + 10)
\][/tex]

4. Conclusion:
- The polynomial [tex]$x^4 + 17x^2 + 70$[/tex] factors completely to:
[tex]\[
(x^2 + 7)(x^2 + 10)
\][/tex]

Therefore, the factors of the polynomial with integer coefficients are [tex]$(x^2 + 7)$[/tex] and [tex]$(x^2 + 10)$[/tex].