The problem asks us to factor the two given expressions: $5x^2 + 10x - 15$ and $3x^4 - 6x^3 + 9x^2$.

1. Problem Description

The problem asks us to factor the two given expressions: $5x^2 + 10x - 15$ and $3x^4 - 6x^3 + 9x^2$.

2. Solution Steps

First expression: $5x^2 + 10x - 15$.

Step 1: Factor out the greatest common factor (GCF), which is

5. $5x^2 + 10x - 15 = 5(x^2 + 2x - 3)$

Step 2: Factor the quadratic expression inside the parenthesis $x^2 + 2x - 3$. We need to find two numbers that multiply to -3 and add up to

2. These numbers are 3 and -

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

Step 3: Substitute the factored quadratic back into the expression.

$5(x^2 + 2x - 3) = 5(x + 3)(x - 1)$

Second expression: $3x^4 - 6x^3 + 9x^2$.

Step 1: Factor out the greatest common factor (GCF), which is $3x^2$.

$3x^4 - 6x^3 + 9x^2 = 3x^2(x^2 - 2x + 3)$

Step 2: Check if the quadratic expression $x^2 - 2x + 3$ can be factored further.

We need to find two numbers that multiply to 3 and add up to -

2. However, no such real numbers exist because the discriminant $b^2 - 4ac = (-2)^2 - 4(1)(3) = 4 - 12 = -8 < 0$. Thus, the quadratic $x^2 - 2x + 3$ cannot be factored using real numbers.

Step 3: The factored expression is $3x^2(x^2 - 2x + 3)$.

3. Final Answer

The factored forms of the expressions are:

$5x^2 + 10x - 15 = 5(x + 3)(x - 1)$

$3x^4 - 6x^3 + 9x^2 = 3x^2(x^2 - 2x + 3)$