Algebra Examples

${(\frac{1}{x + 2})}^{2} + 7 (\frac{1}{x + 2}) + 6$

Step 1

Simplify each term.

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Step 1.1

Apply the product rule to $\frac{1}{x + 2}$ .

$\frac{1^{2}}{{(x + 2)}^{2}} + 7 (\frac{1}{x + 2}) + 6$

Step 1.2

One to any power is one.

$\frac{1}{{(x + 2)}^{2}} + 7 (\frac{1}{x + 2}) + 6$

Step 1.3

Combine $7$ and $\frac{1}{x + 2}$ .

$\frac{1}{{(x + 2)}^{2}} + \frac{7}{x + 2} + 6$

Step 2

Find the common denominator.

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Step 2.1

Multiply $\frac{7}{x + 2}$ by $\frac{x + 2}{x + 2}$ .

$\frac{1}{{(x + 2)}^{2}} + \frac{7}{x + 2} \cdot \frac{x + 2}{x + 2} + 6$

Step 2.2

Multiply $\frac{7}{x + 2}$ by $\frac{x + 2}{x + 2}$ .

$\frac{1}{{(x + 2)}^{2}} + \frac{7 (x + 2)}{(x + 2) (x + 2)} + 6$

Step 2.3

Write $6$ as a fraction with denominator $1$ .

$\frac{1}{{(x + 2)}^{2}} + \frac{7 (x + 2)}{(x + 2) (x + 2)} + \frac{6}{1}$

Step 2.4

Multiply $\frac{6}{1}$ by $\frac{{(x + 2)}^{2}}{{(x + 2)}^{2}}$ .

$\frac{1}{{(x + 2)}^{2}} + \frac{7 (x + 2)}{(x + 2) (x + 2)} + \frac{6}{1} \cdot \frac{{(x + 2)}^{2}}{{(x + 2)}^{2}}$

Step 2.5

Multiply $\frac{6}{1}$ by $\frac{{(x + 2)}^{2}}{{(x + 2)}^{2}}$ .

$\frac{1}{{(x + 2)}^{2}} + \frac{7 (x + 2)}{(x + 2) (x + 2)} + \frac{6 {(x + 2)}^{2}}{{(x + 2)}^{2}}$

Step 2.6

Raise $x + 2$ to the power of $1$ .

$\frac{1}{{(x + 2)}^{2}} + \frac{7 (x + 2)}{{(x + 2)}^{1} (x + 2)} + \frac{6 {(x + 2)}^{2}}{{(x + 2)}^{2}}$

Step 2.7

Raise $x + 2$ to the power of $1$ .

$\frac{1}{{(x + 2)}^{2}} + \frac{7 (x + 2)}{{(x + 2)}^{1} {(x + 2)}^{1}} + \frac{6 {(x + 2)}^{2}}{{(x + 2)}^{2}}$

Step 2.8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{{(x + 2)}^{2}} + \frac{7 (x + 2)}{{(x + 2)}^{1 + 1}} + \frac{6 {(x + 2)}^{2}}{{(x + 2)}^{2}}$

Step 2.9

Add $1$ and $1$ .

$\frac{1}{{(x + 2)}^{2}} + \frac{7 (x + 2)}{{(x + 2)}^{2}} + \frac{6 {(x + 2)}^{2}}{{(x + 2)}^{2}}$

Step 3

Combine the numerators over the common denominator.

$\frac{1 + 7 (x + 2) + 6 {(x + 2)}^{2}}{{(x + 2)}^{2}}$

Step 4

Simplify each term.

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Step 4.1

Apply the distributive property.

$\frac{1 + 7 x + 7 \cdot 2 + 6 {(x + 2)}^{2}}{{(x + 2)}^{2}}$

Step 4.2

Multiply $7$ by $2$ .

$\frac{1 + 7 x + 14 + 6 {(x + 2)}^{2}}{{(x + 2)}^{2}}$

Step 4.3

Rewrite ${(x + 2)}^{2}$ as $(x + 2) (x + 2)$ .

$\frac{1 + 7 x + 14 + 6 ((x + 2) (x + 2))}{{(x + 2)}^{2}}$

Step 4.4

Expand $(x + 2) (x + 2)$ using the FOIL Method.

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Step 4.4.1

Apply the distributive property.

$\frac{1 + 7 x + 14 + 6 (x (x + 2) + 2 (x + 2))}{{(x + 2)}^{2}}$

Step 4.4.2

Apply the distributive property.

$\frac{1 + 7 x + 14 + 6 (x \cdot x + x \cdot 2 + 2 (x + 2))}{{(x + 2)}^{2}}$

Step 4.4.3

Apply the distributive property.

$\frac{1 + 7 x + 14 + 6 (x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2)}{{(x + 2)}^{2}}$

Step 4.5

Simplify and combine like terms.

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Step 4.5.1

Simplify each term.

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Step 4.5.1.1

Multiply $x$ by $x$ .

$\frac{1 + 7 x + 14 + 6 (x^{2} + x \cdot 2 + 2 x + 2 \cdot 2)}{{(x + 2)}^{2}}$

Step 4.5.1.2

Move $2$ to the left of $x$ .

$\frac{1 + 7 x + 14 + 6 (x^{2} + 2 \cdot x + 2 x + 2 \cdot 2)}{{(x + 2)}^{2}}$

Step 4.5.1.3

Multiply $2$ by $2$ .

$\frac{1 + 7 x + 14 + 6 (x^{2} + 2 x + 2 x + 4)}{{(x + 2)}^{2}}$

Step 4.5.2

Add $2 x$ and $2 x$ .

$\frac{1 + 7 x + 14 + 6 (x^{2} + 4 x + 4)}{{(x + 2)}^{2}}$

Step 4.6

Apply the distributive property.

$\frac{1 + 7 x + 14 + 6 x^{2} + 6 (4 x) + 6 \cdot 4}{{(x + 2)}^{2}}$

Step 4.7

Simplify.

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Step 4.7.1

Multiply $4$ by $6$ .

$\frac{1 + 7 x + 14 + 6 x^{2} + 24 x + 6 \cdot 4}{{(x + 2)}^{2}}$

Step 4.7.2

Multiply $6$ by $4$ .

$\frac{1 + 7 x + 14 + 6 x^{2} + 24 x + 24}{{(x + 2)}^{2}}$

Step 5

Simplify by adding terms.

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Step 5.1

Add $1$ and $14$ .

$\frac{7 x + 15 + 6 x^{2} + 24 x + 24}{{(x + 2)}^{2}}$

Step 5.2

Add $7 x$ and $24 x$ .

$\frac{31 x + 15 + 6 x^{2} + 24}{{(x + 2)}^{2}}$

Step 5.3

Simplify the expression.

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Step 5.3.1

Add $15$ and $24$ .

$\frac{31 x + 6 x^{2} + 39}{{(x + 2)}^{2}}$

Step 5.3.2

Reorder $31 x$ and $6 x^{2}$ .

$\frac{6 x^{2} + 31 x + 39}{{(x + 2)}^{2}}$

Step 6

Factor by grouping.

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Step 6.1

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 $a \cdot c = 6 \cdot 39 = 234$ and whose sum is $b = 31$ .

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Step 6.1.1

Factor $31$ out of $31 x$ .

$\frac{6 x^{2} + 31 (x) + 39}{{(x + 2)}^{2}}$

Step 6.1.2

Rewrite $31$ as $13$ plus $18$

$\frac{6 x^{2} + (13 + 18) x + 39}{{(x + 2)}^{2}}$

Step 6.1.3

Apply the distributive property.

$\frac{6 x^{2} + 13 x + 18 x + 39}{{(x + 2)}^{2}}$

Step 6.2

Factor out the greatest common factor from each group.

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Step 6.2.1

Group the first two terms and the last two terms.

$\frac{(6 x^{2} + 13 x) + 18 x + 39}{{(x + 2)}^{2}}$

Step 6.2.2

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

$\frac{x (6 x + 13) + 3 (6 x + 13)}{{(x + 2)}^{2}}$

Step 6.3

Factor the polynomial by factoring out the greatest common factor, $6 x + 13$ .

$\frac{(6 x + 13) (x + 3)}{{(x + 2)}^{2}}$