Question

LOTS OF POINTS Simplify the expression by factoring, showing the steps in your work

Answer 1

Answer:

$\frac{3x+7}{4x+1}$

Step-by-step explanation:

$\frac{12x^2+31x+7}{16x^2+8x+1}$

First you factor $12x^2+31x+7$:

$=\left(12x^2+3x\right)+\left(28x+7\right)$

Then you factor out $3x$ from $12x^2+3x$:

$12x^2+3x$

$=12xx+3x$

$=3\cdot \:4xx+3x$

$=3x\left(4x+1\right)$

Then you factor out 7 from $28x+7$:

$28x+7$

$=7\cdot \:4x+7$

$=7\left(4x+1\right)$

$=3x\left(4x+1\right)+7\left(4x+1\right)$

Factor out common term $4x+1$:

$=\left(4x+1\right)\left(3x+7\right)$

$\frac{\left(4x+1\right)\left(3x+7\right)}{16x^2+8x+1}$

Factor $16x^2+8x+1$:

$=4^2x^2+8x+1$

$=4^2x^2+8x+1^2$

$=\left(4x\right)^2+8x+1^2$

$=\left(4x\right)^2+2\cdot \:4x\cdot \:1+1^2$

Use the perfect square formula:

$=\left(4x+1\right)^2$

$\frac{\left(4x+1\right)\left(3x+7\right)}{\left(4x+1\right)^2}$

Cancel out common factor $4x+1$:

$=\frac{3x+7}{4x+1}$