Solve (x+3)(x+4)/(x+1)(x-1)*x^2(1+x)/x+4*x-1/3(x+3)

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

Evaluate

$$\frac{x^{2}}{3}$$

Short Solution Steps

Use the distributive property to multiply x+3 by x+4 and combine like terms.

$$\frac{x^{2}+7x+12}{\left(x+1\right)\left(x-1\right)}\times \frac{x^{2}\left(1+x\right)}{x+4}\times \frac{x-1}{3\left(x+3\right)}$$

Consider \left(x+1\right)\left(x-1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

$$\frac{x^{2}+7x+12}{x^{2}-1}\times \frac{x^{2}\left(1+x\right)}{x+4}\times \frac{x-1}{3\left(x+3\right)}$$

Use the distributive property to multiply x^{2} by 1+x.

$$\frac{x^{2}+7x+12}{x^{2}-1}\times \frac{x^{2}+x^{3}}{x+4}\times \frac{x-1}{3\left(x+3\right)}$$

Use the distributive property to multiply 3 by x+3.

$$\frac{x^{2}+7x+12}{x^{2}-1}\times \frac{x^{2}+x^{3}}{x+4}\times \frac{x-1}{3x+9}$$

Multiply \frac{x^{2}+7x+12}{x^{2}-1} times \frac{x^{2}+x^{3}}{x+4} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(x^{2}+7x+12\right)\left(x^{2}+x^{3}\right)}{\left(x^{2}-1\right)\left(x+4\right)}\times \frac{x-1}{3x+9}$$

Multiply \frac{\left(x^{2}+7x+12\right)\left(x^{2}+x^{3}\right)}{\left(x^{2}-1\right)\left(x+4\right)} times \frac{x-1}{3x+9} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(x^{2}+7x+12\right)\left(x^{2}+x^{3}\right)\left(x-1\right)}{\left(x^{2}-1\right)\left(x+4\right)\left(3x+9\right)}$$

Factor the expressions that are not already factored.

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

Cancel out \left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right) in both numerator and denominator.

$$\frac{x^{2}}{3}$$

Expand

$$\frac{x^{2}}{3}$$

Short Solution Steps

Use the distributive property to multiply x+3 by x+4 and combine like terms.

$$\frac{x^{2}+7x+12}{\left(x+1\right)\left(x-1\right)}\times \frac{x^{2}\left(1+x\right)}{x+4}\times \frac{x-1}{3\left(x+3\right)}$$

Consider \left(x+1\right)\left(x-1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

$$\frac{x^{2}+7x+12}{x^{2}-1}\times \frac{x^{2}\left(1+x\right)}{x+4}\times \frac{x-1}{3\left(x+3\right)}$$

Use the distributive property to multiply x^{2} by 1+x.

$$\frac{x^{2}+7x+12}{x^{2}-1}\times \frac{x^{2}+x^{3}}{x+4}\times \frac{x-1}{3\left(x+3\right)}$$

Use the distributive property to multiply 3 by x+3.

$$\frac{x^{2}+7x+12}{x^{2}-1}\times \frac{x^{2}+x^{3}}{x+4}\times \frac{x-1}{3x+9}$$

Multiply \frac{x^{2}+7x+12}{x^{2}-1} times \frac{x^{2}+x^{3}}{x+4} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(x^{2}+7x+12\right)\left(x^{2}+x^{3}\right)}{\left(x^{2}-1\right)\left(x+4\right)}\times \frac{x-1}{3x+9}$$

Multiply \frac{\left(x^{2}+7x+12\right)\left(x^{2}+x^{3}\right)}{\left(x^{2}-1\right)\left(x+4\right)} times \frac{x-1}{3x+9} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(x^{2}+7x+12\right)\left(x^{2}+x^{3}\right)\left(x-1\right)}{\left(x^{2}-1\right)\left(x+4\right)\left(3x+9\right)}$$

Factor the expressions that are not already factored.

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

Cancel out \left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right) in both numerator and denominator.

$$\frac{x^{2}}{3}$$