Simplify (2x^2-2)/(3x^2+6x+3)

$\frac{2 x^{2} - 2}{3 x^{2} + 6 x + 3}$

Step 1

Simplify the numerator.

Factor $2$ out of $2 x^{2} - 2$ .

Factor $2$ out of $2 x^{2}$ .

$\frac{2 (x^{2}) - 2}{3 x^{2} + 6 x + 3}$

Factor $2$ out of $- 2$ .

$\frac{2 (x^{2}) + 2 (- 1)}{3 x^{2} + 6 x + 3}$

Factor $2$ out of $2 (x^{2}) + 2 (- 1)$ .

$\frac{2 (x^{2} - 1)}{3 x^{2} + 6 x + 3}$

Rewrite $1$ as $1^{2}$ .

$\frac{2 (x^{2} - 1^{2})}{3 x^{2} + 6 x + 3}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

$\frac{2 (x + 1) (x - 1)}{3 x^{2} + 6 x + 3}$

Step 2

Simplify the denominator.

Factor $3$ out of $3 x^{2} + 6 x + 3$ .

Factor $3$ out of $3 x^{2}$ .

$\frac{2 (x + 1) (x - 1)}{3 (x^{2}) + 6 x + 3}$

Factor $3$ out of $6 x$ .

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

Factor $3$ out of $3$ .

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

Factor $3$ out of $3 (x^{2}) + 3 (2 x)$ .

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

Factor $3$ out of $3 (x^{2} + 2 x) + 3 (1)$ .

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

Factor using the perfect square rule.

Rewrite $1$ as $1^{2}$ .

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

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x = 2 \cdot x \cdot 1$

Rewrite the polynomial.

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

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 1$ .

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

Step 3

Cancel the common factor of $x + 1$ and ${(x + 1)}^{2}$ .

Factor $x + 1$ out of $2 (x + 1) (x - 1)$ .

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

Cancel the common factors.

Factor $x + 1$ out of $3 {(x + 1)}^{2}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify (2x^2-2)/(3x^2+6x+3)

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