Solve for x (4x)/(x+1)=((x+3)(x-1))/(x+1)

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

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$4 x = (x + 3) (x - 1)$

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$(x + 3) (x - 1) = 4 x$

Simplify $(x + 3) (x - 1)$ .

Start simplifying.

$0 + 0 + (x + 3) (x - 1) = 4 x$

Simplify by adding zeros.

$(x + 3) (x - 1) = 4 x$

Expand $(x + 3) (x - 1)$ using the FOIL Method.

Apply the distributive property.

$x (x - 1) + 3 (x - 1) = 4 x$

Apply the distributive property.

$x \cdot x + x \cdot - 1 + 3 (x - 1) = 4 x$

Apply the distributive property.

$x \cdot x + x \cdot - 1 + 3 x + 3 \cdot - 1 = 4 x$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 1 + 3 x + 3 \cdot - 1 = 4 x$

Move $- 1$ to the left of $x$ .

$x^{2} - 1 \cdot x + 3 x + 3 \cdot - 1 = 4 x$

Rewrite $- 1 x$ as $- x$ .

$x^{2} - x + 3 x + 3 \cdot - 1 = 4 x$

Multiply $3$ by $- 1$ .

$x^{2} - x + 3 x - 3 = 4 x$

Add $- x$ and $3 x$ .

$x^{2} + 2 x - 3 = 4 x$

Move all terms containing $x$ to the left side of the equation.

Subtract $4 x$ from both sides of the equation.

$x^{2} + 2 x - 3 - 4 x = 0$

Subtract $4 x$ from $2 x$ .

$x^{2} - 2 x - 3 = 0$

Factor $x^{2} - 2 x - 3$ 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 $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Write the factored form using these integers.

$(x - 3) (x + 1) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 3 = 0$

$x + 1 = 0$

Set $x - 3$ equal to $0$ and solve for $x$ .

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

Set $x + 1$ equal to $0$ and solve for $x$ .

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

The final solution is all the values that make $(x - 3) (x + 1) = 0$ true.

$x = 3, - 1$

Exclude the solutions that do not make $\frac{4 x}{x + 1} = \frac{(x + 3) (x - 1)}{x + 1}$ true.

$x = 3$

Solve for x (4x)/(x+1)=((x+3)(x-1))/(x+1)

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