Pre-Algebra Examples

$(4 x + 5) \div \frac{16 x^{2} - 25}{4 x^{2} + 19 x - 30}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$(4 x + 5) \frac{4 x^{2} + 19 x - 30}{16 x^{2} - 25}$

Step 2

Factor by grouping.

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Step 2.1

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 $a \cdot c = 4 \cdot - 30 = - 120$ and whose sum is $b = 19$ .

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Step 2.1.1

Factor $19$ out of $19 x$ .

$(4 x + 5) \frac{4 x^{2} + 19 (x) - 30}{16 x^{2} - 25}$

Step 2.1.2

Rewrite $19$ as $- 5$ plus $24$

$(4 x + 5) \frac{4 x^{2} + (- 5 + 24) x - 30}{16 x^{2} - 25}$

Step 2.1.3

Apply the distributive property.

$(4 x + 5) \frac{4 x^{2} - 5 x + 24 x - 30}{16 x^{2} - 25}$

Step 2.2

Factor out the greatest common factor from each group.

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Step 2.2.1

Group the first two terms and the last two terms.

$(4 x + 5) \frac{(4 x^{2} - 5 x) + 24 x - 30}{16 x^{2} - 25}$

Step 2.2.2

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

$(4 x + 5) \frac{x (4 x - 5) + 6 (4 x - 5)}{16 x^{2} - 25}$

Step 2.3

Factor the polynomial by factoring out the greatest common factor, $4 x - 5$ .

$(4 x + 5) \frac{(4 x - 5) (x + 6)}{16 x^{2} - 25}$

Step 3

Simplify the denominator.

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Step 3.1

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

$(4 x + 5) \frac{(4 x - 5) (x + 6)}{{(4 x)}^{2} - 25}$

Step 3.2

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

$(4 x + 5) \frac{(4 x - 5) (x + 6)}{{(4 x)}^{2} - 5^{2}}$

Step 3.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 4 x$ and $b = 5$ .

$(4 x + 5) \frac{(4 x - 5) (x + 6)}{(4 x + 5) (4 x - 5)}$

Step 4

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

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Step 4.1

Cancel the common factor.

$(4 x + 5) \frac{(4 x - 5) (x + 6)}{(4 x + 5) (4 x - 5)}$

Step 4.2

Rewrite the expression.

$\frac{(4 x - 5) (x + 6)}{4 x - 5}$

Step 5

Cancel the common factor of $4 x - 5$ .

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Step 5.1

Cancel the common factor.

$\frac{(4 x - 5) (x + 6)}{4 x - 5}$

Step 5.2

Divide $x + 6$ by $1$ .

$x + 6$