Solve C(t)=3t^4+18t^3+15t^2

$$C ( t ) = 3 t ^ { 4 } + 18 t ^ { 3 } + 15 t ^ { 2 }$$

Factor

$$3\left(t+1\right)\left(t+5\right)t^{2}$$

Solution Steps

Factor out 3.

$$3\left(t^{4}+6t^{3}+5t^{2}\right)$$

Consider t^{4}+6t^{3}+5t^{2}. Factor out t^{2}.

$$t^{2}\left(t^{2}+6t+5\right)$$

Consider t^{2}+6t+5. Factor the expression by grouping. First, the expression needs to be rewritten as t^{2}+at+bt+5. To find a and b, set up a system to be solved.

$$a+b=6 ab=1\times 5=5$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. The only such pair is the system solution.

$$a=1 b=5$$

Rewrite t^{2}+6t+5 as \left(t^{2}+t\right)+\left(5t+5\right).

$$\left(t^{2}+t\right)+\left(5t+5\right)$$

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

$$t\left(t+1\right)+5\left(t+1\right)$$

Factor out common term t+1 by using distributive property.

$$\left(t+1\right)\left(t+5\right)$$

Rewrite the complete factored expression.

$$3t^{2}\left(t+1\right)\left(t+5\right)$$

Evaluate

$$3\left(t+1\right)\left(t+5\right)t^{2}$$