Basic Math Examples

$\frac{16 a^{2} + 40 a + 25}{3 a^{2} - 10 a - 8} \div \frac{4 a + 5}{a^{2} - 8 a + 16}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{16 a^{2} + 40 a + 25}{3 a^{2} - 10 a - 8} \cdot \frac{a^{2} - 8 a + 16}{4 a + 5}$

Step 2

Factor using the perfect square rule.

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Step 2.1

Rewrite $16 a^{2}$ as ${(4 a)}^{2}$ .

$\frac{{(4 a)}^{2} + 40 a + 25}{3 a^{2} - 10 a - 8} \cdot \frac{a^{2} - 8 a + 16}{4 a + 5}$

Step 2.2

Rewrite $25$ as $5^{2}$ .

$\frac{{(4 a)}^{2} + 40 a + 5^{2}}{3 a^{2} - 10 a - 8} \cdot \frac{a^{2} - 8 a + 16}{4 a + 5}$

Step 2.3

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

$40 a = 2 \cdot (4 a) \cdot 5$

Step 2.4

Rewrite the polynomial.

$\frac{{(4 a)}^{2} + 2 \cdot (4 a) \cdot 5 + 5^{2}}{3 a^{2} - 10 a - 8} \cdot \frac{a^{2} - 8 a + 16}{4 a + 5}$

Step 2.5

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = 4 a$ and $b = 5$ .

$\frac{{(4 a + 5)}^{2}}{3 a^{2} - 10 a - 8} \cdot \frac{a^{2} - 8 a + 16}{4 a + 5}$

Step 3

Factor by grouping.

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Step 3.1

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 $a \cdot c = 3 \cdot - 8 = - 24$ and whose sum is $b = - 10$ .

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Step 3.1.1

Factor $- 10$ out of $- 10 a$ .

$\frac{{(4 a + 5)}^{2}}{3 a^{2} - 10 (a) - 8} \cdot \frac{a^{2} - 8 a + 16}{4 a + 5}$

Step 3.1.2

Rewrite $- 10$ as $2$ plus $- 12$

$\frac{{(4 a + 5)}^{2}}{3 a^{2} + (2 - 12) a - 8} \cdot \frac{a^{2} - 8 a + 16}{4 a + 5}$

Step 3.1.3

Apply the distributive property.

$\frac{{(4 a + 5)}^{2}}{3 a^{2} + 2 a - 12 a - 8} \cdot \frac{a^{2} - 8 a + 16}{4 a + 5}$

Step 3.2

Factor out the greatest common factor from each group.

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Step 3.2.1

Group the first two terms and the last two terms.

$\frac{{(4 a + 5)}^{2}}{(3 a^{2} + 2 a) - 12 a - 8} \cdot \frac{a^{2} - 8 a + 16}{4 a + 5}$

Step 3.2.2

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

$\frac{{(4 a + 5)}^{2}}{a (3 a + 2) - 4 (3 a + 2)} \cdot \frac{a^{2} - 8 a + 16}{4 a + 5}$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $3 a + 2$ .

$\frac{{(4 a + 5)}^{2}}{(3 a + 2) (a - 4)} \cdot \frac{a^{2} - 8 a + 16}{4 a + 5}$

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $4 a + 5$ .

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Step 4.1.1

Factor $4 a + 5$ out of ${(4 a + 5)}^{2}$ .

$\frac{(4 a + 5) (4 a + 5)}{(3 a + 2) (a - 4)} \cdot \frac{a^{2} - 8 a + 16}{4 a + 5}$

Step 4.1.2

Cancel the common factor.

$\frac{(4 a + 5) (4 a + 5)}{(3 a + 2) (a - 4)} \cdot \frac{a^{2} - 8 a + 16}{4 a + 5}$

Step 4.1.3

Rewrite the expression.

$\frac{4 a + 5}{(3 a + 2) (a - 4)} (a^{2} - 8 a + 16)$

Step 4.2

Multiply $\frac{4 a + 5}{(3 a + 2) (a - 4)}$ by $a^{2} - 8 a + 16$ .

$\frac{(4 a + 5) (a^{2} - 8 a + 16)}{(3 a + 2) (a - 4)}$

Step 5

Factor using the perfect square rule.

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Step 5.1

Rewrite $16$ as $4^{2}$ .

$\frac{(4 a + 5) (a^{2} - 8 a + 4^{2})}{(3 a + 2) (a - 4)}$

Step 5.2

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

$8 a = 2 \cdot a \cdot 4$

Step 5.3

Rewrite the polynomial.

$\frac{(4 a + 5) (a^{2} - 2 \cdot a \cdot 4 + 4^{2})}{(3 a + 2) (a - 4)}$

Step 5.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = a$ and $b = 4$ .

$\frac{(4 a + 5) {(a - 4)}^{2}}{(3 a + 2) (a - 4)}$

Step 6

Cancel the common factor of ${(a - 4)}^{2}$ and $a - 4$ .

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Step 6.1

Factor $a - 4$ out of $(4 a + 5) {(a - 4)}^{2}$ .

$\frac{(a - 4) ((4 a + 5) (a - 4))}{(3 a + 2) (a - 4)}$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $a - 4$ out of $(3 a + 2) (a - 4)$ .

$\frac{(a - 4) ((4 a + 5) (a - 4))}{(a - 4) (3 a + 2)}$

Step 6.2.2

Cancel the common factor.

$\frac{(a - 4) ((4 a + 5) (a - 4))}{(a - 4) (3 a + 2)}$

Step 6.2.3

Rewrite the expression.

$\frac{(4 a + 5) (a - 4)}{3 a + 2}$