Answer:
To solve the inequality 3x^2 + 5x > -2, we can follow these steps:
Move all the terms to the left-hand side to get 3x^2 + 5x + 2 > 0
Factor the quadratic expression by finding two numbers that multiply to 6 (3 times 2) and add to 5. These numbers are 3 and 2, so we can write: 3x^2 + 3x + 2x + 2 > 0
Group the first two terms and the last two terms: 3x(x + 1) + 2(x + 1) > 0
Factor out (x + 1): (x + 1)(3x + 2) > 0
Determine the sign of each factor by testing values of x:
When x < -1, both factors are negative, so the product is positive.
When -1 < x < -2/3, the first factor (x + 1) is positive, but the second factor (3x + 2) is negative, so the product is negative.
When x > -2/3, both factors are positive, so the product is positive.
Write the solution as an inequality based on the sign of the product: (x + 1)(3x + 2) > 0 is true when x < -1 or x > -2/3.
Therefore, the solution to the inequality 3x^2 + 5x > -2 is x < -1 or x > -2/3.
Step-by-step explanation: