Solve -8t^3-8t^2+30t

$$- 8 t ^ { 3 } - 8 t ^ { 2 } + 30 t$$

Factor

$$2t\left(3-2t\right)\left(2t+5\right)$$

Solution Steps

Factor out 2.

$$2\left(-4t^{3}-4t^{2}+15t\right)$$

Consider -4t^{3}-4t^{2}+15t. Factor out t.

$$t\left(-4t^{2}-4t+15\right)$$

Consider -4t^{2}-4t+15. Factor the expression by grouping. First, the expression needs to be rewritten as -4t^{2}+at+bt+15. To find a and b, set up a system to be solved.

$$a+b=-4 ab=-4\times 15=-60$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -60.

$$1,-60 2,-30 3,-20 4,-15 5,-12 6,-10$$

Calculate the sum for each pair.

$$1-60=-59 2-30=-28 3-20=-17 4-15=-11 5-12=-7 6-10=-4$$

The solution is the pair that gives sum -4.

$$a=6 b=-10$$

Rewrite -4t^{2}-4t+15 as \left(-4t^{2}+6t\right)+\left(-10t+15\right).

$$\left(-4t^{2}+6t\right)+\left(-10t+15\right)$$

Factor out 2t in the first and 5 in the second group.

$$2t\left(-2t+3\right)+5\left(-2t+3\right)$$

Factor out common term -2t+3 by using distributive property.

$$\left(-2t+3\right)\left(2t+5\right)$$

Rewrite the complete factored expression.

$$2t\left(-2t+3\right)\left(2t+5\right)$$

Evaluate

$$2t\left(3-2t\right)\left(2t+5\right)$$