Solve left(2a^2+bright)^2-2left(-2a^2right)^2-8{left(1/2bright)}^2+{left(2{a}^2-bright)}^2

$${ \left(2 { a }^{ 2 } +b \right) }^{ 2 } -2 { \left(-2 { a }^{ 2 } \right) }^{ 2 } -8 { \left( \frac{ 1 }{ 2 } b \right) }^{ 2 } + { \left(2 { a }^{ 2 } -b \right) }^{ 2 }$$

Evaluate

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Solution Steps

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(2a^{2}+b\right)^{2}.

$$4\left(a^{2}\right)^{2}+4a^{2}b+b^{2}-2\left(-2a^{2}\right)^{2}-8\times \left(\frac{1}{2}b\right)^{2}+\left(2a^{2}-b\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$4a^{4}+4a^{2}b+b^{2}-2\left(-2a^{2}\right)^{2}-8\times \left(\frac{1}{2}b\right)^{2}+\left(2a^{2}-b\right)^{2}$$

Expand \left(-2a^{2}\right)^{2}.

$$4a^{4}+4a^{2}b+b^{2}-2\left(-2\right)^{2}\left(a^{2}\right)^{2}-8\times \left(\frac{1}{2}b\right)^{2}+\left(2a^{2}-b\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$4a^{4}+4a^{2}b+b^{2}-2\left(-2\right)^{2}a^{4}-8\times \left(\frac{1}{2}b\right)^{2}+\left(2a^{2}-b\right)^{2}$$

Calculate -2 to the power of 2 and get 4.

$$4a^{4}+4a^{2}b+b^{2}-2\times 4a^{4}-8\times \left(\frac{1}{2}b\right)^{2}+\left(2a^{2}-b\right)^{2}$$

Multiply 2 and 4 to get 8.

$$4a^{4}+4a^{2}b+b^{2}-8a^{4}-8\times \left(\frac{1}{2}b\right)^{2}+\left(2a^{2}-b\right)^{2}$$

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

$$-4a^{4}+4a^{2}b+b^{2}-8\times \left(\frac{1}{2}b\right)^{2}+\left(2a^{2}-b\right)^{2}$$

Expand \left(\frac{1}{2}b\right)^{2}.

$$-4a^{4}+4a^{2}b+b^{2}-8\times \left(\frac{1}{2}\right)^{2}b^{2}+\left(2a^{2}-b\right)^{2}$$

Calculate \frac{1}{2} to the power of 2 and get \frac{1}{4}.

$$-4a^{4}+4a^{2}b+b^{2}-8\times \frac{1}{4}b^{2}+\left(2a^{2}-b\right)^{2}$$

Multiply 8 and \frac{1}{4} to get 2.

$$-4a^{4}+4a^{2}b+b^{2}-2b^{2}+\left(2a^{2}-b\right)^{2}$$

Combine b^{2} and -2b^{2} to get -b^{2}.

$$-4a^{4}+4a^{2}b-b^{2}+\left(2a^{2}-b\right)^{2}$$

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(2a^{2}-b\right)^{2}.

$$-4a^{4}+4a^{2}b-b^{2}+4\left(a^{2}\right)^{2}-4a^{2}b+b^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$-4a^{4}+4a^{2}b-b^{2}+4a^{4}-4a^{2}b+b^{2}$$

Combine -4a^{4} and 4a^{4} to get 0.

$$4a^{2}b-b^{2}-4a^{2}b+b^{2}$$

Combine 4a^{2}b and -4a^{2}b to get 0.

$$-b^{2}+b^{2}$$

Combine -b^{2} and b^{2} to get 0.

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Factor

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