Solve f(x)=36x^2+210x+392+2x^3

$$f ( x ) = 36 x ^ { 2 } + 210 x + 392 + 2 x ^ { 3 }$$

Factor

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

Solution Steps

Factor out 2.

$$2\left(18x^{2}+105x+196+x^{3}\right)$$

Consider 18x^{2}+105x+196+x^{3}. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 196 and q divides the leading coefficient 1. One such root is -7. Factor the polynomial by dividing it by x+7.

$$\left(x+7\right)\left(x^{2}+11x+28\right)$$

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.

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

Evaluate

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