Simplify (2x^2-5x-3)/(4x^3+2x^2+2x+1)

$\frac{2 x^{2} - 5 x - 3}{4 x^{3} + 2 x^{2} + 2 x + 1}$

Step 1

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 3 = - 6$ and whose sum is $b = - 5$ .

Factor $- 5$ out of $- 5 x$ .

$\frac{2 x^{2} - 5 (x) - 3}{4 x^{3} + 2 x^{2} + 2 x + 1}$

Rewrite $- 5$ as $1$ plus $- 6$

$\frac{2 x^{2} + (1 - 6) x - 3}{4 x^{3} + 2 x^{2} + 2 x + 1}$

Apply the distributive property.

$\frac{2 x^{2} + 1 x - 6 x - 3}{4 x^{3} + 2 x^{2} + 2 x + 1}$

Multiply $x$ by $1$ .

$\frac{2 x^{2} + x - 6 x - 3}{4 x^{3} + 2 x^{2} + 2 x + 1}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(2 x^{2} + x) - 6 x - 3}{4 x^{3} + 2 x^{2} + 2 x + 1}$

Factor out the greatest common factor (GCF) from each group.

$\frac{x (2 x + 1) - 3 (2 x + 1)}{4 x^{3} + 2 x^{2} + 2 x + 1}$

Factor the polynomial by factoring out the greatest common factor, $2 x + 1$ .

$\frac{(2 x + 1) (x - 3)}{4 x^{3} + 2 x^{2} + 2 x + 1}$

Step 2

Simplify the denominator.

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(2 x + 1) (x - 3)}{(4 x^{3} + 2 x^{2}) + 2 x + 1}$

Factor out the greatest common factor (GCF) from each group.

$\frac{(2 x + 1) (x - 3)}{2 x^{2} (2 x + 1) + 1 (2 x + 1)}$

Factor the polynomial by factoring out the greatest common factor, $2 x + 1$ .

$\frac{(2 x + 1) (x - 3)}{(2 x + 1) (2 x^{2} + 1)}$

Step 3

Cancel the common factor of $2 x + 1$ .

Cancel the common factor.

$\frac{(2 x + 1) (x - 3)}{(2 x + 1) (2 x^{2} + 1)}$

Rewrite the expression.

$\frac{x - 3}{2 x^{2} + 1}$

Simplify (2x^2-5x-3)/(4x^3+2x^2+2x+1)

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