We are asked to factor the quadratic expressions. Problem 15: $4y^2 - 23y + 15$ Problem 18: $10x^2 + 9x + 2$ Problem 21: $12x^2 + 4x - 5$

1. Problem Description

We are asked to factor the quadratic expressions.

Problem 15: $4y^2 - 23y + 15$

Problem 18: $10x^2 + 9x + 2$

Problem 21: $12x^2 + 4x - 5$

2. Solution Steps

Problem 15: $4y^2 - 23y + 15$

We are looking for two numbers that multiply to $4 * 15 = 60$ and add up to $-23$. These numbers are $-20$ and $-3$.

$4y^2 - 20y - 3y + 15$

$4y(y - 5) - 3(y - 5)$

$(4y - 3)(y - 5)$

Problem 18: $10x^2 + 9x + 2$

We are looking for two numbers that multiply to $10 * 2 = 20$ and add up to $9$. These numbers are $5$ and $4$.

$10x^2 + 5x + 4x + 2$

$5x(2x + 1) + 2(2x + 1)$

$(5x + 2)(2x + 1)$

Problem 21: $12x^2 + 4x - 5$

We are looking for two numbers that multiply to $12 * -5 = -60$ and add up to $4$. These numbers are $10$ and $-6$.

$12x^2 + 10x - 6x - 5$

$2x(6x + 5) - 1(6x + 5)$

$(2x - 1)(6x + 5)$

3. Final Answer

Problem 15: $(4y - 3)(y - 5)$

Problem 18: $(5x + 2)(2x + 1)$

Problem 21: $(2x - 1)(6x + 5)$