Find the domain over which the curve y=x^3+12x^2 +45x -30 is increasing

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The function y=x^3+12x^2+45x-30 is increasing in the set where y' is positive.

y' = 3x^2 + 24x + 45

3x^2 + 24x + 45 > 0

=> x^2 + 8x + 15 > 0

=> x^2 + 5x + 3x + 15 > 0

=> x(x + 5) + 3(x...

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The function y=x^3+12x^2+45x-30 is increasing in the set where y' is positive.

y' = 3x^2 + 24x + 45

3x^2 + 24x + 45 > 0

=> x^2 + 8x + 15 > 0

=> x^2 + 5x + 3x + 15 > 0

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

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

This is true if:

(x + 3) > 0 and (x + 5) > 0

=> x > -3 and x > -5

=> x > -3

(x + 3) < 0 and (x + 5) < 0

=> x < -3 and x < -5

=> x < -5

The set of values of x for which the function is increasing is `(-oo, -5)U(-3, oo)`

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