First factor the each term in the expression.

44x³ =  2 · 2 · 11 · x · x · x

24x² =  2 · 2 · 2 · 3 · x · x

The GCF is the product of the common prime factors in each term.

GCF = 2 · 2 · x · x = 4x² 

Write each term as a product of the GCF and its remaining factors.

44x³  +24x² = 4x² (11x) + 4x² (6)

Combine like terms uisng distribuitve property: ab + ac = a(b + c).

44x³  +24x² = 4x² (11x + 6)

Simplified expression is 4x² (11x + 6).