We are asked to factor three quadratic expressions: 1. $2x^2 + 7x + 3$

1. Problem Description

We are asked to factor three quadratic expressions:

1. $2x^2 + 7x + 3$

2. $3x^2 + 8x + 4$

3. $2x^2 - x - 15$

2. Solution Steps

1. Factor $2x^2 + 7x + 3$:

We look for two numbers that multiply to $2 \cdot 3 = 6$ and add to $7$. The numbers are $6$ and $1$.

Rewrite the middle term: $2x^2 + 6x + 1x + 3$.

Factor by grouping: $2x(x + 3) + 1(x + 3)$.

Factor out the common term $(x + 3)$: $(2x + 1)(x + 3)$.

2. Factor $3x^2 + 8x + 4$:

We look for two numbers that multiply to $3 \cdot 4 = 12$ and add to $8$. The numbers are $6$ and $2$.

Rewrite the middle term: $3x^2 + 6x + 2x + 4$.

Factor by grouping: $3x(x + 2) + 2(x + 2)$.

Factor out the common term $(x + 2)$: $(3x + 2)(x + 2)$.

3. Factor $2x^2 - x - 15$:

We look for two numbers that multiply to $2 \cdot -15 = -30$ and add to $-1$. The numbers are $-6$ and $5$.

Rewrite the middle term: $2x^2 - 6x + 5x - 15$.

Factor by grouping: $2x(x - 3) + 5(x - 3)$.

Factor out the common term $(x - 3)$: $(2x + 5)(x - 3)$.

3. Final Answer

1. $2x^2 + 7x + 3 = (2x + 1)(x + 3)$

2. $3x^2 + 8x + 4 = (3x + 2)(x + 2)$

3. $2x^2 - x - 15 = (2x + 5)(x - 3)$