Solve z^2+z+4z^3+4z^2=

$$z ^ { 2 } + z + 4 z ^ { 3 } + 4 z ^ { 2 } =$$

Factor

$$z\left(z+1\right)\left(4z+1\right)$$

Solution Steps

Factor out z.

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

Consider z+1+4z^{2}+4z. Multiply and combine like terms.

$$4z^{2}+5z+1$$

Consider 4z^{2}+5z+1. Factor the expression by grouping. First, the expression needs to be rewritten as 4z^{2}+az+bz+1. To find a and b, set up a system to be solved.

$$a+b=5 ab=4\times 1=4$$

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 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 5.

$$a=1 b=4$$

Rewrite 4z^{2}+5z+1 as \left(4z^{2}+z\right)+\left(4z+1\right).

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

Factor out z in 4z^{2}+z.

$$z\left(4z+1\right)+4z+1$$

Factor out common term 4z+1 by using distributive property.

$$\left(4z+1\right)\left(z+1\right)$$

Rewrite the complete factored expression.

$$z\left(4z+1\right)\left(z+1\right)$$

Evaluate

$$z\left(z+1\right)\left(4z+1\right)$$

Solution Steps

Combine z^{2} and 4z^{2} to get 5z^{2}.

$$5z^{2}+z+4z^{3}$$