Solve -18h-9+72h^2+36h+72h+36

$$- 18 h - 9 + 72 h ^ { 2 } + 36 h + 72 h + 36$$

Evaluate

$$72h^{2}+90h+27$$

Solution Steps

Combine -18h and 36h to get 18h.

$$18h-9+72h^{2}+72h+36$$

Combine 18h and 72h to get 90h.

$$90h-9+72h^{2}+36$$

Add -9 and 36 to get 27.

$$90h+27+72h^{2}$$

Factor

$$9\left(2h+1\right)\left(4h+3\right)$$

Solution Steps

Factor out 9.

$$9\left(10h+3+8h^{2}\right)$$

Consider -2h-1+8h^{2}+4h+8h+4. Multiply and combine like terms.

$$8h^{2}+10h+3$$

Consider 8h^{2}+10h+3. Factor the expression by grouping. First, the expression needs to be rewritten as 8h^{2}+ah+bh+3. To find a and b, set up a system to be solved.

$$a+b=10 ab=8\times 3=24$$

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 24.

$$1,24 2,12 3,8 4,6$$

Calculate the sum for each pair.

$$1+24=25 2+12=14 3+8=11 4+6=10$$

The solution is the pair that gives sum 10.

$$a=4 b=6$$

Rewrite 8h^{2}+10h+3 as \left(8h^{2}+4h\right)+\left(6h+3\right).

$$\left(8h^{2}+4h\right)+\left(6h+3\right)$$

Factor out 4h in the first and 3 in the second group.

$$4h\left(2h+1\right)+3\left(2h+1\right)$$

Factor out common term 2h+1 by using distributive property.

$$\left(2h+1\right)\left(4h+3\right)$$

Rewrite the complete factored expression.

$$9\left(2h+1\right)\left(4h+3\right)$$