1. ## Factoring

2. $\displaystyle 24w^2-71w-30=(3w-10)\cdot(8w+3)$
It's a Quadratic equation, so using the quadratic formula for solving the equation $\displaystyle 24w^2-71w-30=0$: $\displaystyle w_{1,2}=\frac {71\pm\sqrt{71^2-4\cdot 24\cdot (-30)}}{2\cdot 24}$ which gives the solutions $\displaystyle \frac {10}3$ and $\displaystyle -\frac{3}{8}$.

3. Hello, jennifer_person!

Factor: .$\displaystyle 24w^2 -71w-30$
What does that mean?
Surely, it's easier to say "find an answer that is correct"
. . (as opposed to several billion answers that are not correct).

Has anyone taught you the way to factor trinomials (in general)?

We have: .$\displaystyle 24w^2 - 71w - 30$

Multiply the first coeffient by the last coefficient: .$\displaystyle 24\cdot30 \:=\:720$

Note the sign of the last term.
. . If it is "plus", think "SUM".
. . If it is "minus", think "DIFFERENCE".

The last term is -30, so we think "difference."

Factor 720 into two parts whose difference is the middle coefficient, 71.

How do we find that pair?
Start dividing 720 by 1, 2, 3, 4, etc., keeping the ones that "come out even."

$\displaystyle 720 \;=\;\begin{array}{cc}1\cdot720 \\ 2\cdot360 \\ 3\cdot240 \\ 4\cdot180 \\ 5\cdot144 \\ 6\cdot120 \\ 8\cdot90 \\ 9\cdot 80 & \Leftarrow \text{ Here! a difference of 71}\end{array}$

The pair we want is: (9, 80).

Since the middle term is -71, we will use: +9 and -80
We use these numbers as coefficients for the "middle term".

We have: .$\displaystyle 24w^2+ 9x - 80w - 30$

Factor in pairs: .$\displaystyle 3w\underbrace{(8w+3)} - 10\underbrace{(8w + 3)}$
. . . . . . . . . . . . . . . . .$\displaystyle ^{\text{common factor}}$

Factor: . $\displaystyle (8w+3)(3w-10)$ . . . There!