Solve 3x^4+33x^3+84x^2

$$3 x ^ { 4 } + 33 x ^ { 3 } + 84 x ^ { 2 }$$

Factor

$$3\left(x+4\right)\left(x+7\right)x^{2}$$

Solution Steps

Factor out 3.

$$3\left(x^{4}+11x^{3}+28x^{2}\right)$$

Consider x^{4}+11x^{3}+28x^{2}. Factor out x^{2}.

$$x^{2}\left(x^{2}+11x+28\right)$$

Consider x^{2}+11x+28. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+28. To find a and b, set up a system to be solved.

$$a+b=11 ab=1\times 28=28$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 28.

$$1,28 2,14 4,7$$

Calculate the sum for each pair.

$$1+28=29 2+14=16 4+7=11$$

The solution is the pair that gives sum 11.

$$a=4 b=7$$

Rewrite x^{2}+11x+28 as \left(x^{2}+4x\right)+\left(7x+28\right).

$$\left(x^{2}+4x\right)+\left(7x+28\right)$$

Factor out x in the first and 7 in the second group.

$$x\left(x+4\right)+7\left(x+4\right)$$

Factor out common term x+4 by using distributive property.

$$\left(x+4\right)\left(x+7\right)$$

Rewrite the complete factored expression.

$$3x^{2}\left(x+4\right)\left(x+7\right)$$

Evaluate

$$3\left(x+4\right)\left(x+7\right)x^{2}$$