We need to factorize the following quadratic expressions: 5) $4v^2 - 31v - 8$ 6) $10v^2 - 39v + 27$ 7) $8r^2 + 12r$ 8) $8m^2 - 6m$

1. Problem Description

We need to factorize the following quadratic expressions:

5) $4v^2 - 31v - 8$

6) $10v^2 - 39v + 27$

7) $8r^2 + 12r$

8) $8m^2 - 6m$

2. Solution Steps

5) To factorize $4v^2 - 31v - 8$, we are looking for two numbers that multiply to $4 \times -8 = -32$ and add up to $-31$. These numbers are $-32$ and $1$. We can rewrite the middle term as $-32v + v$.

$4v^2 - 31v - 8 = 4v^2 - 32v + v - 8$

$= 4v(v - 8) + 1(v - 8)$

$= (4v + 1)(v - 8)$

6) To factorize $10v^2 - 39v + 27$, we are looking for two numbers that multiply to $10 \times 27 = 270$ and add up to $-39$. These numbers are $-30$ and $-9$. We can rewrite the middle term as $-30v - 9v$.

$10v^2 - 39v + 27 = 10v^2 - 30v - 9v + 27$

$= 10v(v - 3) - 9(v - 3)$

$= (10v - 9)(v - 3)$

7) To factorize $8r^2 + 12r$, we can take out the greatest common factor, which is $4r$.

$8r^2 + 12r = 4r(2r + 3)$

8) To factorize $8m^2 - 6m$, we can take out the greatest common factor, which is $2m$.

$8m^2 - 6m = 2m(4m - 3)$

3. Final Answer

5) $(4v + 1)(v - 8)$

6) $(10v - 9)(v - 3)$

7) $4r(2r + 3)$

8) $2m(4m - 3)$