Solve -3a+12a^2-12a^3

$$- 3 a + 12 a ^ { 2 } - 12 a ^ { 3 }$$

Factor

$$-3a\left(1-2a\right)^{2}$$

Solution Steps

Factor out 3.

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

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

$$a\left(-1+4a-4a^{2}\right)$$

Consider -1+4a-4a^{2}. Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-4a^{2}+4a-1$$

Factor the expression by grouping. First, the expression needs to be rewritten as -4a^{2}+pa+qa-1. To find p and q, set up a system to be solved.

$$p+q=4 pq=-4\left(-1\right)=4$$

Since pq is positive, p and q have the same sign. Since p+q is positive, p and q are both positive. List all such integer pairs that give product 4.

$$1,4 2,2$$

Calculate the sum for each pair.

$$1+4=5 2+2=4$$

The solution is the pair that gives sum 4.

$$p=2 q=2$$

Rewrite -4a^{2}+4a-1 as \left(-4a^{2}+2a\right)+\left(2a-1\right).

$$\left(-4a^{2}+2a\right)+\left(2a-1\right)$$

Factor out -2a in -4a^{2}+2a.

$$-2a\left(2a-1\right)+2a-1$$

Factor out common term 2a-1 by using distributive property.

$$\left(2a-1\right)\left(-2a+1\right)$$

Rewrite the complete factored expression.

$$3a\left(2a-1\right)\left(-2a+1\right)$$

Evaluate

$$-3a\left(1-2a\right)^{2}$$