# Divide ((4k^2-3k-1)/(4k^2+11k+7))/((16k^2-1)/(4k^2+3k-7))

4k2-3k-14k2+11k+716k2-14k2+3k-7
Multiply the numerator by the reciprocal of the denominator.
4k2-3k-14k2+11k+7⋅4k2+3k-716k2-1
Factor by grouping.
For a polynomial of the form ax2+bx+c, rewrite the middle term as a sum of two terms whose product is a⋅c=4⋅-1=-4 and whose sum is b=-3.
Factor -3 out of -3k.
4k2-3(k)-14k2+11k+7⋅4k2+3k-716k2-1
Rewrite -3 as 1 plus -4
4k2+(1-4)k-14k2+11k+7⋅4k2+3k-716k2-1
Apply the distributive property.
4k2+1k-4k-14k2+11k+7⋅4k2+3k-716k2-1
Multiply k by 1.
4k2+k-4k-14k2+11k+7⋅4k2+3k-716k2-1
4k2+k-4k-14k2+11k+7⋅4k2+3k-716k2-1
Factor out the greatest common factor from each group.
Group the first two terms and the last two terms.
(4k2+k)-4k-14k2+11k+7⋅4k2+3k-716k2-1
Factor out the greatest common factor (GCF) from each group.
k(4k+1)-(4k+1)4k2+11k+7⋅4k2+3k-716k2-1
k(4k+1)-(4k+1)4k2+11k+7⋅4k2+3k-716k2-1
Factor the polynomial by factoring out the greatest common factor, 4k+1.
(4k+1)(k-1)4k2+11k+7⋅4k2+3k-716k2-1
(4k+1)(k-1)4k2+11k+7⋅4k2+3k-716k2-1
Factor by grouping.
For a polynomial of the form ax2+bx+c, rewrite the middle term as a sum of two terms whose product is a⋅c=4⋅7=28 and whose sum is b=11.
Factor 11 out of 11k.
(4k+1)(k-1)4k2+11(k)+7⋅4k2+3k-716k2-1
Rewrite 11 as 4 plus 7
(4k+1)(k-1)4k2+(4+7)k+7⋅4k2+3k-716k2-1
Apply the distributive property.
(4k+1)(k-1)4k2+4k+7k+7⋅4k2+3k-716k2-1
(4k+1)(k-1)4k2+4k+7k+7⋅4k2+3k-716k2-1
Factor out the greatest common factor from each group.
Group the first two terms and the last two terms.
(4k+1)(k-1)(4k2+4k)+7k+7⋅4k2+3k-716k2-1
Factor out the greatest common factor (GCF) from each group.
(4k+1)(k-1)4k(k+1)+7(k+1)⋅4k2+3k-716k2-1
(4k+1)(k-1)4k(k+1)+7(k+1)⋅4k2+3k-716k2-1
Factor the polynomial by factoring out the greatest common factor, k+1.
(4k+1)(k-1)(k+1)(4k+7)⋅4k2+3k-716k2-1
(4k+1)(k-1)(k+1)(4k+7)⋅4k2+3k-716k2-1
Factor by grouping.
For a polynomial of the form ax2+bx+c, rewrite the middle term as a sum of two terms whose product is a⋅c=4⋅-7=-28 and whose sum is b=3.
Factor 3 out of 3k.
(4k+1)(k-1)(k+1)(4k+7)⋅4k2+3(k)-716k2-1
Rewrite 3 as -4 plus 7
(4k+1)(k-1)(k+1)(4k+7)⋅4k2+(-4+7)k-716k2-1
Apply the distributive property.
(4k+1)(k-1)(k+1)(4k+7)⋅4k2-4k+7k-716k2-1
(4k+1)(k-1)(k+1)(4k+7)⋅4k2-4k+7k-716k2-1
Factor out the greatest common factor from each group.
Group the first two terms and the last two terms.
(4k+1)(k-1)(k+1)(4k+7)⋅(4k2-4k)+7k-716k2-1
Factor out the greatest common factor (GCF) from each group.
(4k+1)(k-1)(k+1)(4k+7)⋅4k(k-1)+7(k-1)16k2-1
(4k+1)(k-1)(k+1)(4k+7)⋅4k(k-1)+7(k-1)16k2-1
Factor the polynomial by factoring out the greatest common factor, k-1.
(4k+1)(k-1)(k+1)(4k+7)⋅(k-1)(4k+7)16k2-1
(4k+1)(k-1)(k+1)(4k+7)⋅(k-1)(4k+7)16k2-1
Simplify the denominator.
Rewrite 16k2 as (4k)2.
(4k+1)(k-1)(k+1)(4k+7)⋅(k-1)(4k+7)(4k)2-1
Rewrite 1 as 12.
(4k+1)(k-1)(k+1)(4k+7)⋅(k-1)(4k+7)(4k)2-12
Since both terms are perfect squares, factor using the difference of squares formula, a2-b2=(a+b)(a-b) where a=4k and b=1.
(4k+1)(k-1)(k+1)(4k+7)⋅(k-1)(4k+7)(4k+1)(4k-1)
(4k+1)(k-1)(k+1)(4k+7)⋅(k-1)(4k+7)(4k+1)(4k-1)
Cancel the common factor of 4k+1.
Cancel the common factor.
(4k+1)(k-1)(k+1)(4k+7)⋅(k-1)(4k+7)(4k+1)(4k-1)
Rewrite the expression.
k-1(k+1)(4k+7)⋅(k-1)(4k+7)4k-1
k-1(k+1)(4k+7)⋅(k-1)(4k+7)4k-1
Cancel the common factor of 4k+7.
Factor 4k+7 out of (k+1)(4k+7).
k-1(4k+7)(k+1)⋅(k-1)(4k+7)4k-1
Factor 4k+7 out of (k-1)(4k+7).
k-1(4k+7)(k+1)⋅(4k+7)(k-1)4k-1
Cancel the common factor.
k-1(4k+7)(k+1)⋅(4k+7)(k-1)4k-1
Rewrite the expression.
k-1k+1⋅k-14k-1
k-1k+1⋅k-14k-1
Multiply k-1k+1 and k-14k-1.
(k-1)(k-1)(k+1)(4k-1)
Raise k-1 to the power of 1.
(k-1)1(k-1)(k+1)(4k-1)
Raise k-1 to the power of 1.
(k-1)1(k-1)1(k+1)(4k-1)
Use the power rule aman=am+n to combine exponents.
(k-1)1+1(k+1)(4k-1)