DC CAPE Algebra I Seeing Structure in Expressions. Practice it free.

The greatest common factor of $20x^{2}$ and $30x$ is $10x\textsf{.}$ Factoring gives $10x(2x - 3)$.

$4x^{2} - 9 = (2x)^{2} - 3^{2} = (2x - 3)(2x + 3)$.

Write $30u$ with denominator $g$: $\dfrac{10u}{g} + \dfrac{30ug}{g} = \dfrac{10u + 30ug}{g} = \dfrac{10u(1 + 3g)}{g}$.

Find two numbers with product $-12$ and sum $1$: $4$ and $-3$. So $x^2 + x - 12 = (x+4)(x-3)$.

$x^3 - 8 = (x - 2)(x^2 + 2x + 4) = 0$. The quadratic factor has no real roots, so the only real root is $x = 2$.

$x^{2} - 16$ is a difference of squares: $x^{2} - 4^{2}$. So $x^{2} - 16 = (x + 4)(x - 4)$.

Divide both sides by $2$: $x^{2} + 4x = 3$.

The greatest common factor of $16x^{3}$, $24x^{2}$, and $8x$ is $8x\textsf{.}$ Factoring gives $8x(2x^{2} + 3x - 1)$.