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------------------------------------------------ Factor the expression completely.

[tex]\[ 70x^4 + 80x^2 \] [/tex]

To factor the expression [tex]$70x^4 + 80x^2$[/tex] completely, let's break it down into simpler steps:

1. Identify the Greatest Common Factor (GCF):

First, identify the common factor shared by the terms. Both terms, [tex]$70x^4$[/tex] and [tex]$80x^2$[/tex], can be divided by [tex]$10x^2$[/tex].

- The GCF of the numerical coefficients 70 and 80 is 10.
- The smallest power of [tex]$x$[/tex] common to both terms is [tex]$x^2$[/tex].

Therefore, the GCF of the entire expression is [tex]$10x^2$[/tex].

2. Factor out the GCF:

Divide each term by the GCF and place the GCF outside the parentheses:

[tex]\[
70x^4 + 80x^2 = 10x^2(7x^2) + 10x^2(8)
\][/tex]

Simplifying inside the parentheses, we get:

[tex]\[
10x^2(7x^2 + 8)
\][/tex]

3. Check for Further Factoring:

Now, we look inside the parentheses to see if [tex]$7x^2 + 8$[/tex] can be factored further. Both terms 7 and 8 are constants with no common factors other than 1, and [tex]$7x^2$[/tex] and 8 are not perfect squares or cubes, so this part is already fully factored.

Therefore, the completely factored form of the expression [tex]$70x^4 + 80x^2$[/tex] is:

[tex]\[
10x^2(7x^2 + 8)
\][/tex]

This is the final answer.