The problem asks us to factor each of the given quadratic expressions completely. The given expressions are: 1) $3x^2 - 4x - 7$ 2) $3n^2 - 13n + 14$ 3) $2a^2 + 11a + 9$ 4) $5b^2 - 49b + 72$

1. Problem Description

The problem asks us to factor each of the given quadratic expressions completely. The given expressions are:

1) $3x^2 - 4x - 7$

2) $3n^2 - 13n + 14$

3) $2a^2 + 11a + 9$

4) $5b^2 - 49b + 72$

2. Solution Steps

1) $3x^2 - 4x - 7$

We are looking for two numbers that multiply to $3*(-7) = -21$ and add to $-4$. Those numbers are $-7$ and $3$. We rewrite the middle term as $-7x + 3x$:

$3x^2 - 7x + 3x - 7$

Now, we factor by grouping:

$x(3x - 7) + 1(3x - 7)$

$(x+1)(3x-7)$

2) $3n^2 - 13n + 14$

We are looking for two numbers that multiply to $3*14 = 42$ and add to $-13$. Those numbers are $-6$ and $-7$. We rewrite the middle term as $-6n - 7n$:

$3n^2 - 6n - 7n + 14$

Now, we factor by grouping:

$3n(n - 2) - 7(n - 2)$

$(3n - 7)(n - 2)$

3) $2a^2 + 11a + 9$

We are looking for two numbers that multiply to $2*9 = 18$ and add to $11$. Those numbers are $2$ and $9$. We rewrite the middle term as $2a + 9a$:

$2a^2 + 2a + 9a + 9$

Now, we factor by grouping:

$2a(a + 1) + 9(a + 1)$

$(2a + 9)(a + 1)$

4) $5b^2 - 49b + 72$

We are looking for two numbers that multiply to $5*72 = 360$ and add to $-49$. Those numbers are $-40$ and $-9$. We rewrite the middle term as $-40b - 9b$:

$5b^2 - 40b - 9b + 72$

Now, we factor by grouping:

$5b(b - 8) - 9(b - 8)$

$(5b - 9)(b - 8)$

3. Final Answer

1) $(x+1)(3x-7)$

2) $(3n-7)(n-2)$

3) $(2a+9)(a+1)$

4) $(5b-9)(b-8)$