Solve (4)(a^2+9)^2-(a+3)(3-a)(a^2+9)

$$( 4 ) ( a ^ { 2 } + 9 ) ^ { 2 } - ( a + 3 ) ( 3 - a ) ( a ^ { 2 } + 9 )$$

Evaluate

$$\left(5a^{2}+27\right)\left(a^{2}+9\right)$$

Solution Steps

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(a^{2}+9\right)^{2}.

$$4\left(\left(a^{2}\right)^{2}+18a^{2}+81\right)-\left(a+3\right)\left(3-a\right)\left(a^{2}+9\right)$$

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

$$4\left(a^{4}+18a^{2}+81\right)-\left(a+3\right)\left(3-a\right)\left(a^{2}+9\right)$$

Use the distributive property to multiply 4 by a^{4}+18a^{2}+81.

$$4a^{4}+72a^{2}+324-\left(a+3\right)\left(3-a\right)\left(a^{2}+9\right)$$

Use the distributive property to multiply a+3 by 3-a and combine like terms.

$$4a^{4}+72a^{2}+324-\left(-a^{2}+9\right)\left(a^{2}+9\right)$$

Use the distributive property to multiply -a^{2}+9 by a^{2}+9 and combine like terms.

$$4a^{4}+72a^{2}+324-\left(-a^{4}+81\right)$$

To find the opposite of -a^{4}+81, find the opposite of each term.

$$4a^{4}+72a^{2}+324+a^{4}-81$$

Combine 4a^{4} and a^{4} to get 5a^{4}.

$$5a^{4}+72a^{2}+324-81$$

Subtract 81 from 324 to get 243.

$$5a^{4}+72a^{2}+243$$

Expand

$$5a^{4}+72a^{2}+243$$

Solution Steps

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(a^{2}+9\right)^{2}.

$$4\left(\left(a^{2}\right)^{2}+18a^{2}+81\right)-\left(a+3\right)\left(3-a\right)\left(a^{2}+9\right)$$

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

$$4\left(a^{4}+18a^{2}+81\right)-\left(a+3\right)\left(3-a\right)\left(a^{2}+9\right)$$

Use the distributive property to multiply 4 by a^{4}+18a^{2}+81.

$$4a^{4}+72a^{2}+324-\left(a+3\right)\left(3-a\right)\left(a^{2}+9\right)$$

Use the distributive property to multiply a+3 by 3-a and combine like terms.

$$4a^{4}+72a^{2}+324-\left(-a^{2}+9\right)\left(a^{2}+9\right)$$

Use the distributive property to multiply -a^{2}+9 by a^{2}+9 and combine like terms.

$$4a^{4}+72a^{2}+324-\left(-a^{4}+81\right)$$

To find the opposite of -a^{4}+81, find the opposite of each term.

$$4a^{4}+72a^{2}+324+a^{4}-81$$

Combine 4a^{4} and a^{4} to get 5a^{4}.

$$5a^{4}+72a^{2}+324-81$$

Subtract 81 from 324 to get 243.

$$5a^{4}+72a^{2}+243$$