Lutasin ang f(x)=(x^2+4x+3/x^2+2x-3-x^2+2x+1/x^2+3x+2)*x+2/x+1

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

I-evaluate

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

Maiikling Hakbang sa Solution

I-factor ang mga expression na hindi pa na-factor sa \frac{x^{2}+4x+3}{x^{2}+2x-3}.

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

I-cancel out ang x+3 sa parehong numerator at denominator.

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

I-factor ang mga expression na hindi pa na-factor sa \frac{x^{2}+2x+1}{x^{2}+3x+2}.

$$\left(\frac{x+1}{x-1}-\frac{\left(x+1\right)^{2}}{\left(x+1\right)\left(x+2\right)}\right)\times \frac{x+2}{x+1}$$

I-cancel out ang x+1 sa parehong numerator at denominator.

$$\left(\frac{x+1}{x-1}-\frac{x+1}{x+2}\right)\times \frac{x+2}{x+1}$$

Para magdagdag o mag-subtract ng mga expression, i-expand ang mga iyon para gawing magkakapareho ang mga denominator ng mga ito. Ang least common multiple ng x-1 at x+2 ay \left(x-1\right)\left(x+2\right). I-multiply ang \frac{x+1}{x-1} times \frac{x+2}{x+2}. I-multiply ang \frac{x+1}{x+2} times \frac{x-1}{x-1}.

$$\left(\frac{\left(x+1\right)\left(x+2\right)}{\left(x-1\right)\left(x+2\right)}-\frac{\left(x+1\right)\left(x-1\right)}{\left(x-1\right)\left(x+2\right)}\right)\times \frac{x+2}{x+1}$$

Dahil may parehong denominator ang \frac{\left(x+1\right)\left(x+2\right)}{\left(x-1\right)\left(x+2\right)} at \frac{\left(x+1\right)\left(x-1\right)}{\left(x-1\right)\left(x+2\right)}, ibawas ang mga ito sa pamamagitan ng pagbawas sa mga numerator ng mga ito.

$$\frac{\left(x+1\right)\left(x+2\right)-\left(x+1\right)\left(x-1\right)}{\left(x-1\right)\left(x+2\right)}\times \frac{x+2}{x+1}$$

Gawin ang mga pag-multiply sa \left(x+1\right)\left(x+2\right)-\left(x+1\right)\left(x-1\right).

$$\frac{x^{2}+2x+x+2-x^{2}+x-x+1}{\left(x-1\right)\left(x+2\right)}\times \frac{x+2}{x+1}$$

Pagsamahin ang magkakatulad na term sa x^{2}+2x+x+2-x^{2}+x-x+1.

$$\frac{3x+3}{\left(x-1\right)\left(x+2\right)}\times \frac{x+2}{x+1}$$

I-multiply ang \frac{3x+3}{\left(x-1\right)\left(x+2\right)} sa \frac{x+2}{x+1} sa pamamagitan ng pag-multiply ng numerator sa numerator at denominator sa denominator.

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

I-cancel out ang x+2 sa parehong numerator at denominator.

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

I-factor ang mga expression na hindi pa na-factor.

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

I-cancel out ang x+1 sa parehong numerator at denominator.

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

Palawakin

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

Maiikling Hakbang sa Solution

I-factor ang mga expression na hindi pa na-factor sa \frac{x^{2}+4x+3}{x^{2}+2x-3}.

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

I-cancel out ang x+3 sa parehong numerator at denominator.

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

I-factor ang mga expression na hindi pa na-factor sa \frac{x^{2}+2x+1}{x^{2}+3x+2}.

$$\left(\frac{x+1}{x-1}-\frac{\left(x+1\right)^{2}}{\left(x+1\right)\left(x+2\right)}\right)\times \frac{x+2}{x+1}$$

I-cancel out ang x+1 sa parehong numerator at denominator.

$$\left(\frac{x+1}{x-1}-\frac{x+1}{x+2}\right)\times \frac{x+2}{x+1}$$

Para magdagdag o mag-subtract ng mga expression, i-expand ang mga iyon para gawing magkakapareho ang mga denominator ng mga ito. Ang least common multiple ng x-1 at x+2 ay \left(x-1\right)\left(x+2\right). I-multiply ang \frac{x+1}{x-1} times \frac{x+2}{x+2}. I-multiply ang \frac{x+1}{x+2} times \frac{x-1}{x-1}.

$$\left(\frac{\left(x+1\right)\left(x+2\right)}{\left(x-1\right)\left(x+2\right)}-\frac{\left(x+1\right)\left(x-1\right)}{\left(x-1\right)\left(x+2\right)}\right)\times \frac{x+2}{x+1}$$

Dahil may parehong denominator ang \frac{\left(x+1\right)\left(x+2\right)}{\left(x-1\right)\left(x+2\right)} at \frac{\left(x+1\right)\left(x-1\right)}{\left(x-1\right)\left(x+2\right)}, ibawas ang mga ito sa pamamagitan ng pagbawas sa mga numerator ng mga ito.

$$\frac{\left(x+1\right)\left(x+2\right)-\left(x+1\right)\left(x-1\right)}{\left(x-1\right)\left(x+2\right)}\times \frac{x+2}{x+1}$$

Gawin ang mga pag-multiply sa \left(x+1\right)\left(x+2\right)-\left(x+1\right)\left(x-1\right).

$$\frac{x^{2}+2x+x+2-x^{2}+x-x+1}{\left(x-1\right)\left(x+2\right)}\times \frac{x+2}{x+1}$$

Pagsamahin ang magkakatulad na term sa x^{2}+2x+x+2-x^{2}+x-x+1.

$$\frac{3x+3}{\left(x-1\right)\left(x+2\right)}\times \frac{x+2}{x+1}$$

I-multiply ang \frac{3x+3}{\left(x-1\right)\left(x+2\right)} sa \frac{x+2}{x+1} sa pamamagitan ng pag-multiply ng numerator sa numerator at denominator sa denominator.

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

I-cancel out ang x+2 sa parehong numerator at denominator.

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

I-factor ang mga expression na hindi pa na-factor.

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

I-cancel out ang x+1 sa parehong numerator at denominator.

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