Question 4 Factor completely: \( 40 x^{4}-60 x^{3}+100 x^{2} \) (A) \( 5 x\left(8 x^{3}-12 x^{2}+20 x\right) \) (B) \( 20 x^{2}\left(2 x^{2}-3 x+5\right) \) (C) \( 20\left(2 x^{4}-3 x^{3}+5 x^{2}\right) \) (D) \( 10 x^{2}\left(4 x^{2}-6 x+10\right) \)

To factor the expression \( 40 x^{4}-60 x^{3}+100 x^{2} \) completely, let’s first take out the greatest common factor (GCF). The GCF of the coefficients \( 40, 60, \) and \( 100 \) is \( 20 \), and the lowest degree of \( x \) is \( x^2 \). So, we factor out \( 20 x^{2} \): \[ 20 x^{2}(2 x^{2} - 3 x + 5) \] Now, looking inside the parentheses, \( 2 x^{2} - 3 x + 5 \) does not factor neatly into real numbers, as the discriminant \( b^{2} - 4ac \) yields a negative result. Hence the expression is fully factored as: \[ 20 x^{2}(2 x^{2} - 3 x + 5) \] The correct answer is (B) \( 20 x^{2}\left(2 x^{2}-3 x+5\right) \). Moreover, let's sprinkle in some fun facts! Factoring polynomials like this is a key concept in algebra, with roots tracing back to ancient civilizations like the Babylonians, who developed early forms of algebraic equations. These ancient mathematicians laid the groundwork for methods we still use today! To avoid pitfalls when factoring polynomials, always look for a GCF first! Many students jump right into complex factoring techniques without checking for the simple things they can factor out, leading to unnecessary confusion and mistakes. Always simplify first!