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

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

Combine the numerators over the common denominator.

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

Simplify each term.

Apply the distributive property.

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

Multiply $- 1$ by $- 4$ .

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

Simplify by adding terms.

Subtract $x$ from $2 x$ .

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

Add $- 3$ and $4$ .

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

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 $a \cdot c = 4 \cdot 1 = 4$ and whose sum is $b = 5$ .

Factor $5$ out of $5 x$ .

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

Rewrite $5$ as $1$ plus $4$

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

Apply the distributive property.

$\frac{x + 1}{4 x^{2} + 1 x + 4 x + 1}$

Multiply $x$ by $1$ .

$\frac{x + 1}{4 x^{2} + x + 4 x + 1}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

$\frac{x + 1}{x (4 x + 1) + 1 (4 x + 1)}$

Factor the polynomial by factoring out the greatest common factor, $4 x + 1$ .

$\frac{x + 1}{(4 x + 1) (x + 1)}$

Cancel the common factor of $x + 1$ .

Cancel the common factor.

$\frac{x + 1}{(4 x + 1) (x + 1)}$

Rewrite the expression.

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

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

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