Question

In the following exercises, factor each expression using any method$24n^2+20n+4$

Short answer

The factors are$4\left(2n+1\right)\left(3n+1\right)$

Step-by-step solution

Step 1. Given Information

The trinomial is$24n^2+20n+4$

Find factors using 'ac' method.

Step 2. Find GCF 

Find GCF and factor it

GCF is 4

So, trinomial becomes$4\left(6n^2+5n+1\right)$

Step 3. Find 'ac'  

Compare the trinomial inside the parenthesis with $a{x}^2+bx+c$

So, $a=6,b=5,c=1$

so, $ac=6$

Find two number such as their product is 6 and sum is 5

So numbers are 2,3.

Step 4. Split the middle and factor trinomial by grouping  

Split the middle term

$4\left(6n^2+3n+2n+1\right)$

Factor it

$$\begin{gathered} 4\left(3n\left(2n+1\right)+1\left(2n+1\right)\right) \\ 4\left(2n+1\right)\left(3n+1\right) \end{gathered}$$

Step 5. Check

Multiply the factors $4\left(2n+1\right)\left(3n+1\right)$

$$\begin{gathered} 4\left(6n^2+2n+3n+1\right) \\ 4\left(6n^2+5n+1\right) \\ 24n^2+20n+4 \end{gathered}$$