When we factor numbers, we look for the largest number that divides into all of them. Transitioning this skill to algebraic expressions works the exact same way, except now we also have variables to consider. To find the greatest common factor of a polynomial, we look at the coefficients and the variable terms separately.

Let's break down the expression $8x^4 - 4x^3 + 10x^2$. First, we look at the numbers: 8, -4, and 10. Their greatest common numerical factor is 2. Next, we look at the variables: $x^4$, $x^3$, and $x^2$. The highest power of $x$ that we can divide out of all three terms is $x^2$. Combining these gives us a total GCF of $2x^2$.

When we divide $2x^2$ out of each term, we are left with the simplified polynomial $4x^2 - 2x + 5$. Written in factored form, the original expression becomes:

$8x^4 - 4x^3 + 10x^2 = 2x^2(4x^2 - 2x + 5)$

We can always double-check our work by distributing the GCF back through the parentheses. If multiplying it back out gives us the original expression, we know our factoring is correct. Let's try finding a GCF together.