We want to completely factor the given expression. To do so, we will first identify and factor out the greatest common factor.

Factor Out the GCF

The greatest common factor (GCF) of an expression is a common factor of the terms in the expression. It is the common factor with the greatest coefficient and the greatest exponent. The GCF of the given expression is

3 r .

30 r^{3} + 51 r^{2} + 9 r

SplitIntoFactors

Split into factors

3 r (10 r^{2}) + 3 r (17 r) + 3 r (3)

FactorOut

Factor out $3 r$

3 r (10 r^{2} + 17 r + 3)

Factor the Quadratic Trinomial

Here we have a quadratic trinomial of the form

a r^{2} + b r + c,

where

∣ a ∣ \neq = 1

and there are no common factors. To factor this expression, we will rewrite the middle term,

b r,

as two terms. The coefficients of these two terms will be factors of

a c

whose sum must be

b .

3 r (10 r^{2} + 17 r + 3) \Leftrightarrow 3 r (10 r^{2} + 17 r + 3)

We have that

a = 10,

b = 17,

and

c = 3 .

There are now three steps we need to follow in order to rewrite the above expression.

Find $a c .$ Since we have that $a = 10$ and $c = 3,$ the value of $a c$ is $10 \times 3 = 30 .$

Find factors of $a c .$ Since $a c = 30,$ which is positive, we need factors of $a c$ to have the same sign — both positive or both negative — in order for the product to be positive. Since $b = 17,$ which is also positive, those factors will need to be positive so that their sum is positive.

\underline{1^{st} Factor} 235 \underline{2^{nd} Factor} 15106 \underline{Sum} 2 + 15 3 + 10 5 + 6 \underline{Result} 171311

Rewrite $b x$ as two terms. Now that we know which factors are the ones to be used, we can rewrite $b x$ as two terms.
$3 r (10 r^{2} + 17 r + 3) ⇕ 3 r (10 r^{2} + 2 x + 15 r + 3)$

Finally, we will factor the last expression obtained.

3 r (10 r^{2} + 2 r + 15 r + 3)

FactorOut

Factor out $2 r$

3 r (2 r (5 r + 1) + 15 r + 3)

FactorOut

Factor out $3$

3 r (2 r (5 r + 1) + 3 (5 r + 1))

FactorOut

Factor out $(5 r + 1)$

3 r (5 r + 1) (2 r + 3)

Checking Our Answer

Check your answer

✓

We can expand our answer and compare it with the given expression.

3 r (2 r + 3) (5 r + 1)

Distr

Distribute $3 r$

(6 r^{2} + 9 r) (5 r + 1)

Distr

Distribute $6 r^{2} + 9 r$

5 r (6 r^{2} + 9 r) + (6 r^{2} + 9 r)

Distr

Distribute $5 r$

30 r^{3} + 45 r^{2} + (6 r^{2} + 9 r)

Distr

Distribute $1$

30 r^{3} + 45 r^{2} + 6 r^{2} + 9 r

AddTerms

Add terms

30 r^{3} + 51 r^{2} + 9 r

We can see above that after expanding and simplifying, the result is the same as the given expression. Therefore, we can be sure our solution is correct!

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