Precalculus Examples

${(3 x + 2)}^{2} + 8 (3 x + 2) + 12$

Step 1

Let $u = 3 x + 2$ . Substitute $u$ for all occurrences of $3 x + 2$ .

$u^{2} + 8 u + 12$

Step 2

Factor $u^{2} + 8 u + 12$ using the AC method.

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Step 2.1

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 $12$ and whose sum is $8$ .

$2, 6$

Step 2.2

Write the factored form using these integers.

$(u + 2) (u + 6)$

Step 3

Replace all occurrences of $u$ with $3 x + 2$ .

$(3 x + 2 + 2) (3 x + 2 + 6)$

Step 4

Simplify.

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Step 4.1

Add $2$ and $2$ .

$(3 x + 4) (3 x + 2 + 6)$

Step 4.2

Add $2$ and $6$ .

$(3 x + 4) (3 x + 8)$