Question

which of the following expressions is equivalent to $18x^4 - 12x^2$?
$6\left(3x^2 - 2x^2\
ight)$
$6x^2\left(3x^2 - 2\
ight)$
$x^2\left(18x^2 - 12\
ight)$
$6x\left(2x^3 - 2x\
ight)$

Explanation:

Step1: Find GCF of coefficients and variables

For \(18x^4 - 12x^2\), GCF of 18 and 12 is 6, GCF of \(x^4\) and \(x^2\) is \(x^2\). So GCF is \(6x^2\).

Step2: Factor out GCF

Factor \(6x^2\) from \(18x^4 - 12x^2\): \(6x^2(3x^2 - 2)\).

Step3: Check other options

• Yellow: \(6(3x^2 - 2x^2)=6(x^2)=6x^2

eq18x^4 - 12x^2\).

• Orange: \(x^2(18x^2 - 12)=18x^4 - 12x^2\), but wait, GCF is \(6x^2\), not \(x^2\) (we can factor 6 from \(18x^2 - 12\) too, so \(x^2\times6(3x^2 - 2)=6x^2(3x^2 - 2)\), but the option is \(x^2(18x^2 - 12)\), which is also equivalent but the purple one is factored with GCF properly. Wait, no, let's expand purple: \(6x^2(3x^2 - 2)=18x^4 - 12x^2\), orange: \(x^2(18x^2 - 12)=18x^4 - 12x^2\), teal: \(6x(2x^3 - 2x)=12x^4 - 12x^2

eq18x^4 - 12x^2\). Wait, but the purple one is factoring out the full GCF (6x²), while orange is factoring out x² first. But the question is equivalent. Wait, maybe I made a mistake. Wait, let's re - check.

Wait, the original expression is \(18x^4-12x^2\). Let's expand each option:

• Yellow: \(6(3x^2 - 2x^2)=6(x^2)=6x^2

eq18x^4 - 12x^2\).

• Purple: \(6x^2(3x^2 - 2)=18x^4-12x^2\) (correct).
• Orange: \(x^2(18x^2 - 12)=18x^4 - 12x^2\) (also correct? Wait, but \(18x^2 - 12 = 6(3x^2 - 2)\), so \(x^2\times6(3x^2 - 2)=6x^2(3x^2 - 2)\), same as purple. But maybe the question considers factoring out the greatest common factor. The GCF of 18 and 12 is 6, GCF of \(x^4\) and \(x^2\) is \(x^2\), so GCF is \(6x^2\). So factoring out \(6x^2\) gives \(6x^2(3x^2 - 2)\), which is the purple option. The orange option factors out \(x^2\) but not the full numerical GCF (since 18 and 12 have a GCF of 6). So the purple option is factored with the greatest common factor, so it's the most appropriate equivalent expression when factoring completely.

Answer:

B. \(6x^2(3x^2 - 2)\) (the purple option, assuming the options are labeled as Yellow: \(6(3x^2 - 2x^2)\), Purple: \(6x^2(3x^2 - 2)\), Orange: \(x^2(18x^2 - 12)\), Teal: \(6x(2x^3 - 2x)\))