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

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

Factor $x^{2} - 5 x + 4$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $4$ and whose sum is $- 5$ .

$- 4, - 1$

Write the factored form using these integers.

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

Factor by grouping.

Reorder terms.

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

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 $a \cdot c = - 1 \cdot 4 = - 4$ and whose sum is $b = 3$ .

Factor $3$ out of $3 x$ .

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

Rewrite $3$ as $- 1$ plus $4$

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

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

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

Cancel the common factor of $x - 4$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{x - 1}{- x - 1}$

Factor $- 1$ out of $- x$ .

$\frac{x - 1}{- (x) - 1}$

Rewrite $- 1$ as $- 1 (1)$ .

$\frac{x - 1}{- (x) - 1 (1)}$

Factor $- 1$ out of $- (x) - 1 (1)$ .

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

Rewrite negatives.

Rewrite $- (x + 1)$ as $- 1 (x + 1)$ .

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

Move the negative in front of the fraction.

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

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

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