Simplify (6p^2-13p+5)/(2p^2+17p-9)*(p^2+16p+63)/(4p+28)

$\frac{6 p^{2} - 13 p + 5}{2 p^{2} + 17 p - 9} \cdot \frac{p^{2} + 16 p + 63}{4 p + 28}$

Step 1

Factor by grouping.

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 $a \cdot c = 6 \cdot 5 = 30$ and whose sum is $b = - 13$ .

Factor $- 13$ out of $- 13 p$ .

$\frac{6 p^{2} - 13 (p) + 5}{2 p^{2} + 17 p - 9} \cdot \frac{p^{2} + 16 p + 63}{4 p + 28}$

Rewrite $- 13$ as $- 3$ plus $- 10$

$\frac{6 p^{2} + (- 3 - 10) p + 5}{2 p^{2} + 17 p - 9} \cdot \frac{p^{2} + 16 p + 63}{4 p + 28}$

Apply the distributive property.

$\frac{6 p^{2} - 3 p - 10 p + 5}{2 p^{2} + 17 p - 9} \cdot \frac{p^{2} + 16 p + 63}{4 p + 28}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(6 p^{2} - 3 p) - 10 p + 5}{2 p^{2} + 17 p - 9} \cdot \frac{p^{2} + 16 p + 63}{4 p + 28}$

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

$\frac{3 p (2 p - 1) - 5 (2 p - 1)}{2 p^{2} + 17 p - 9} \cdot \frac{p^{2} + 16 p + 63}{4 p + 28}$

Factor the polynomial by factoring out the greatest common factor, $2 p - 1$ .

$\frac{(2 p - 1) (3 p - 5)}{2 p^{2} + 17 p - 9} \cdot \frac{p^{2} + 16 p + 63}{4 p + 28}$

Step 2

Factor by grouping.

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 $a \cdot c = 2 \cdot - 9 = - 18$ and whose sum is $b = 17$ .

Factor $17$ out of $17 p$ .

$\frac{(2 p - 1) (3 p - 5)}{2 p^{2} + 17 (p) - 9} \cdot \frac{p^{2} + 16 p + 63}{4 p + 28}$

Rewrite $17$ as $- 1$ plus $18$

$\frac{(2 p - 1) (3 p - 5)}{2 p^{2} + (- 1 + 18) p - 9} \cdot \frac{p^{2} + 16 p + 63}{4 p + 28}$

Apply the distributive property.

$\frac{(2 p - 1) (3 p - 5)}{2 p^{2} - 1 p + 18 p - 9} \cdot \frac{p^{2} + 16 p + 63}{4 p + 28}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(2 p - 1) (3 p - 5)}{(2 p^{2} - 1 p) + 18 p - 9} \cdot \frac{p^{2} + 16 p + 63}{4 p + 28}$

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

$\frac{(2 p - 1) (3 p - 5)}{p (2 p - 1) + 9 (2 p - 1)} \cdot \frac{p^{2} + 16 p + 63}{4 p + 28}$

Factor the polynomial by factoring out the greatest common factor, $2 p - 1$ .

$\frac{(2 p - 1) (3 p - 5)}{(2 p - 1) (p + 9)} \cdot \frac{p^{2} + 16 p + 63}{4 p + 28}$

Step 3

Factor $p^{2} + 16 p + 63$ using the AC method.

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 $63$ and whose sum is $16$ .

$7, 9$

Write the factored form using these integers.

$\frac{(2 p - 1) (3 p - 5)}{(2 p - 1) (p + 9)} \cdot \frac{(p + 7) (p + 9)}{4 p + 28}$

Step 4

Simplify terms.

Factor $4$ out of $4 p + 28$ .

Factor $4$ out of $4 p$ .

$\frac{(2 p - 1) (3 p - 5)}{(2 p - 1) (p + 9)} \cdot \frac{(p + 7) (p + 9)}{4 (p) + 28}$

Factor $4$ out of $28$ .

$\frac{(2 p - 1) (3 p - 5)}{(2 p - 1) (p + 9)} \cdot \frac{(p + 7) (p + 9)}{4 p + 4 \cdot 7}$

Factor $4$ out of $4 p + 4 \cdot 7$ .

$\frac{(2 p - 1) (3 p - 5)}{(2 p - 1) (p + 9)} \cdot \frac{(p + 7) (p + 9)}{4 (p + 7)}$

Cancel the common factor of $p + 9$ .

Factor $p + 9$ out of $(2 p - 1) (p + 9)$ .

$\frac{(2 p - 1) (3 p - 5)}{(p + 9) (2 p - 1)} \cdot \frac{(p + 7) (p + 9)}{4 (p + 7)}$

Factor $p + 9$ out of $(p + 7) (p + 9)$ .

$\frac{(2 p - 1) (3 p - 5)}{(p + 9) (2 p - 1)} \cdot \frac{(p + 9) (p + 7)}{4 (p + 7)}$

Cancel the common factor.

$\frac{(2 p - 1) (3 p - 5)}{(p + 9) (2 p - 1)} \cdot \frac{(p + 9) (p + 7)}{4 (p + 7)}$

Rewrite the expression.

$\frac{(2 p - 1) (3 p - 5)}{2 p - 1} \cdot \frac{p + 7}{4 (p + 7)}$

Multiply $\frac{(2 p - 1) (3 p - 5)}{2 p - 1}$ by $\frac{p + 7}{4 (p + 7)}$ .

$\frac{(2 p - 1) (3 p - 5) (p + 7)}{(2 p - 1) (4 (p + 7))}$

Cancel the common factor of $2 p - 1$ .

Cancel the common factor.

$\frac{(2 p - 1) (3 p - 5) (p + 7)}{(2 p - 1) (4 (p + 7))}$

Rewrite the expression.

$\frac{(3 p - 5) (p + 7)}{4 (p + 7)}$

Cancel the common factor of $p + 7$ .

Cancel the common factor.

$\frac{(3 p - 5) (p + 7)}{4 (p + 7)}$

Rewrite the expression.

$\frac{3 p - 5}{4}$

Simplify (6p^2-13p+5)/(2p^2+17p-9)*(p^2+16p+63)/(4p+28)

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