Simplify (x^2+4x+4)/(2x^2+3x-2)

$\frac{x^{2} + 4 x + 4}{2 x^{2} + 3 x - 2}$

Step 1

Factor using the perfect square rule.

Rewrite $4$ as $2^{2}$ .

$\frac{x^{2} + 4 x + 2^{2}}{2 x^{2} + 3 x - 2}$

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

$4 x = 2 \cdot x \cdot 2$

Rewrite the polynomial.

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

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

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

Step 2

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 2 = - 4$ and whose sum is $b = 3$ .

Factor $3$ out of $3 x$ .

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

Rewrite $3$ as $- 1$ plus $4$

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

Factor out the greatest common factor (GCF) from each group.

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

Factor the polynomial by factoring out the greatest common factor, $2 x - 1$ .

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

Step 3

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

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

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{x + 2}{2 x - 1}$

Simplify (x^2+4x+4)/(2x^2+3x-2)

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