Given:
[tex]$\frac{12 x^{2}+31 x+7}{16 x^{2}+8 x+1}[/tex]
To find:
The simplified expression by factoring.
Solution:
Let us factor the numerator:
[tex]12 x^{2}+31 x+7[/tex]
31x can be written as 3x + 28x.
[tex]=\left(12 x^{2}+3 x\right)+(28 x+7)[/tex]
Take 3x common in 1st two terms and 7 common in next two terms.
[tex]=3 x(4 x+1)+7(4 x+1)[/tex]
Make sure the remaining terms in the both brackets must be same.
Now, take out common factor (4x + 1).
[tex]=(4 x+1)(3 x+7)[/tex]
In the same way, factorize the denominator:
[tex]16 x^{2}+8 x+1[/tex]
8x can be written as 4x + 4x.
[tex]=(16 x^{2}+4 x)+(4x+1)[/tex]
Take 4x common in 1st two terms and 1 common in next two terms.
[tex]=4x(4 x+1)+1(4x+1)[/tex]
Make sure the remaining terms in the both brackets must be same.
Now, take out common factor (4x + 1).
[tex]=(4x+1)(4x+1)[/tex]
Substitute the terms we found for numerator and denominator:
[tex]$\frac{12 x^{2}+31 x+7}{16 x^{2}+8 x+1}=\frac{(4 x+1)(3 x+7)}{(4 x+1)(4 x+1)}[/tex]
Cancel the common factors.
[tex]$=\frac{3 x+7}{4 x+1}[/tex]
The simplified expression is [tex]\frac{3 x+7}{4 x+1}[/tex].
