Answer:
[tex]\frac{3x+7}{4x+1}[/tex]
Step-by-step explanation:
[tex]\frac{12x^2+31x+7}{16x^2+8x+1}[/tex]
First you factor [tex]12x^2+31x+7[/tex]:
[tex]=\left(12x^2+3x\right)+\left(28x+7\right)[/tex]
Then you factor out [tex]3x[/tex] from [tex]12x^2+3x[/tex]:
[tex]12x^2+3x[/tex]
[tex]=12xx+3x[/tex]
[tex]=3\cdot \:4xx+3x[/tex]
[tex]=3x\left(4x+1\right)[/tex]
Then you factor out 7 from [tex]28x+7[/tex]:
[tex]28x+7[/tex]
[tex]=7\cdot \:4x+7[/tex]
[tex]=7\left(4x+1\right)[/tex]
[tex]=3x\left(4x+1\right)+7\left(4x+1\right)[/tex]
Factor out common term [tex]4x+1[/tex]:
[tex]=\left(4x+1\right)\left(3x+7\right)[/tex]
[tex]\frac{\left(4x+1\right)\left(3x+7\right)}{16x^2+8x+1}[/tex]
Factor [tex]16x^2+8x+1[/tex]:
[tex]=4^2x^2+8x+1[/tex]
[tex]=4^2x^2+8x+1^2[/tex]
[tex]=\left(4x\right)^2+8x+1^2[/tex]
[tex]=\left(4x\right)^2+2\cdot \:4x\cdot \:1+1^2[/tex]
Use the perfect square formula:
[tex]=\left(4x+1\right)^2[/tex]
[tex]\frac{\left(4x+1\right)\left(3x+7\right)}{\left(4x+1\right)^2}[/tex]
Cancel out common factor [tex]4x+1[/tex]:
[tex]=\frac{3x+7}{4x+1}[/tex]
