Solve -8p^{4}+12p^3+4p^4-8p^2+3p^2

$$- 8 p ^ { 4 } + 12 p ^ { 3 } + 4 p ^ { 4 } - 8 p ^ { 2 } + 3 p ^ { 2 }$$

Evaluate

$$\left(1-2p\right)\left(2p-5\right)p^{2}$$

Solution Steps

Combine -8p^{4} and 4p^{4} to get -4p^{4}.

$$-4p^{4}+12p^{3}-8p^{2}+3p^{2}$$

Combine -8p^{2} and 3p^{2} to get -5p^{2}.

$$-4p^{4}+12p^{3}-5p^{2}$$

Factor

$$\left(1-2p\right)\left(2p-5\right)p^{2}$$

Solution Steps

Factor out p^{2}.

$$p^{2}\left(-4p^{2}+12p-5\right)$$

Consider -8p^{2}+12p+4p^{2}-8+3. Multiply and combine like terms.

$$-4p^{2}+12p-5$$

Consider -4p^{2}+12p-5. Factor the expression by grouping. First, the expression needs to be rewritten as -4p^{2}+ap+bp-5. To find a and b, set up a system to be solved.

$$a+b=12 ab=-4\left(-5\right)=20$$

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 20.

$$1,20 2,10 4,5$$

Calculate the sum for each pair.

$$1+20=21 2+10=12 4+5=9$$

The solution is the pair that gives sum 12.

$$a=10 b=2$$

Rewrite -4p^{2}+12p-5 as \left(-4p^{2}+10p\right)+\left(2p-5\right).

$$\left(-4p^{2}+10p\right)+\left(2p-5\right)$$

Factor out -2p in -4p^{2}+10p.

$$-2p\left(2p-5\right)+2p-5$$

Factor out common term 2p-5 by using distributive property.

$$\left(2p-5\right)\left(-2p+1\right)$$

Rewrite the complete factored expression.

$$p^{2}\left(2p-5\right)\left(-2p+1\right)$$