Solve the inequality 10x^3+6x^2-90x-54<0 and write your answer using interval notation.

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We have to solve the inequality: 10x^3+6x^2-90x-54<0

10x^3 + 6x^2 - 90x - 54 < 0

=> 2x^2(5x + 3) - 18(5x + 3) < 0

=> (2x^2 - 18)(5x + 3) < 0

=> (x - 3)(x + 3)(5x + 3) < 0

For the polynomial to be true either only one of the terms should be less than 0 or all the terms should be less than 0.

1. (x - 3)<0, (x + 3)>0 and (5x + 3)>0

=> x < 3, x > -3 and x > -3/5

The values of x that satisfy all the conditions is (-3/5, 3)

2. (x - 3)>0, (x + 3)<0 and (5x + 3)>0

=> x > 3, x < -3 and x > -3/5

No value of x satisfies all the conditions.

3. (x - 3)>0, (x + 3)>0 and (5x + 3)<0

=> x > 3, x < -3 and x < -3/5

No value of x satisfies all the conditions.

4. (x - 3)<0, (x + 3)<0 and (5x + 3)<0

=> x < 3, x < -3 and x < -3/5

The values of x that satisfy all the conditions is (-inf., -3)

The required solution of the inequality is (-inf, -3)U(-3/5, 3).

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