Simplify (x+1)/(x-4)-(x+1)/(x^2-7x+12)

$\frac{x + 1}{x - 4} - \frac{x + 1}{x^{2} - 7 x + 12}$

Factor $x^{2} - 7 x + 12$ 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 $12$ and whose sum is $- 7$ .

$- 4, - 3$

Write the factored form using these integers.

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

To write $\frac{x + 1}{x - 4}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

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

Simplify terms.

Multiply $\frac{x + 1}{x - 4}$ by $\frac{x - 3}{x - 3}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Factor $x + 1$ out of $(x + 1) (x - 3) - (x + 1)$ .

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

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

Factor $x + 1$ out of $(x + 1) (x - 3) + (x + 1) \cdot - 1$ .

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

Subtract $1$ from $- 3$ .

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

Cancel the common factor of $x - 4$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify (x+1)/(x-4)-(x+1)/(x^2-7x+12)

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