To factorize the expression \(8a + 125ax^3\), we can start by finding the greatest common factor (GCF) of the two terms. The GCF of \(8a\) and \(125ax^3\) is \(a\), so we can factor that out:

\(8a + 125ax^3 = a(8 + 125x^2)\)

Now, let's focus on factorizing the expression \(8 + 125x^2\). This is a sum of cubes because \(8\) is \(2^3\) and \(125x^2\) is \((5x)^3\). The sum of cubes formula is:

\(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)

In our case, \(a = 2\) and \(b = 5x\), so we have:

\(8 + 125x^2 = (2 + 5x)(4 - 10x + 25x^2)\)

Now, let's put it all together:

\(8a + 125ax^3 = a(8 + 125x^2) = a(2 + 5x)(4 - 10x + 25x^2)\)