Question

Factor completely:

$6q^2-9q-6$

Short answer

The factored form is $3(q-2)(2q+1)$.

Step-by-step solution

Step 1. Given Information 

We are given a polynomial $6q^2-9q-6$,

We have to factor the given polynomial.

Step 2. Factoring the polynomial   

Factoring out GCF $3$, we get

$6q^2-9q-6=3\left(2q^2-3q-2\right)$

Using middle terms splitting or acmethod, we get

$6q^2-9q-6=3\left(2q^2-4q+q-2\right)$

Grouping the terms, we get

$$\begin{gathered} 6q^2-9q-6=3\left(\left(2q^2-4q\right)+(q-2)\right) \\ 6q^2-9q-6=3\left(2q\right(q-2)+1(q-2\left)\right) \end{gathered}$$

Factoring out GCF $q-2$, we get

$6q^2-9q-6=3(q-2)(2q+1)$

Step 3. Checking the factors   

Multiplying the factors, we get

$$\begin{gathered} 3(q-2)(2q+1)=6q^2-9q-6 \\ 3\left(2q\right(q-2)+1(q-2\left)\right)=6q^2-9q-6 \\ 3\left(2q^2-4q+q-2\right)=6q^2-9q-6 \\ 3\left(2q^2-3q-2\right)=6q^2-9q-6 \\ 6q^2-9q-6=6q^2-9q-6 \\ LHS=RHS \end{gathered}$$

Hence, the factors are correct.