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

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

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

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

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.

(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

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)

Add 1 and 1.

(k-1)2(k+1)(4k-1)

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