Solve {l}{2(a+3)^{2}-7(a+4)(a-4)+4(-a-1)^{2}}{-2(a^2+9a^2+2)-7(a^2-4^2)+4(a+1)}

$$\left. \begin{array} { l } { 2 ( a + 3 ) ^ { 2 } - 7 ( a + 4 ) ( a - 4 ) + 4 ( - a - 1 ) ^ { 2 } } \\ { - 2 ( a ^ { 2 } + 9 a ^ { 2 } + 2 ) - 7 ( a ^ { 2 } - 4 ^ { 2 } ) + 4 ( a + 1 ) } \end{array} \right.$$

Evaluate

$$246+24a-28a^{2}$$

Solution Steps

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(a+3\right)^{2}.

$$2\left(a^{2}+6a+9\right)-7\left(a+4\right)\left(a-4\right)+4\left(-a-1\right)^{2}-2\left(a^{2}+9a^{2}+2\right)-7\left(a^{2}-4^{2}\right)+4\left(a+1\right)$$

Use the distributive property to multiply 2 by a^{2}+6a+9.

$$2a^{2}+12a+18-7\left(a+4\right)\left(a-4\right)+4\left(-a-1\right)^{2}-2\left(a^{2}+9a^{2}+2\right)-7\left(a^{2}-4^{2}\right)+4\left(a+1\right)$$

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(-a-1\right)^{2}.

$$2a^{2}+12a+18-7\left(a+4\right)\left(a-4\right)+4\left(\left(-a\right)^{2}-2\left(-a\right)+1\right)-2\left(a^{2}+9a^{2}+2\right)-7\left(a^{2}-4^{2}\right)+4\left(a+1\right)$$

Calculate -a to the power of 2 and get a^{2}.

$$2a^{2}+12a+18-7\left(a+4\right)\left(a-4\right)+4\left(a^{2}-2\left(-a\right)+1\right)-2\left(a^{2}+9a^{2}+2\right)-7\left(a^{2}-4^{2}\right)+4\left(a+1\right)$$

Multiply -2 and -1 to get 2.

$$2a^{2}+12a+18-7\left(a+4\right)\left(a-4\right)+4\left(a^{2}+2a+1\right)-2\left(a^{2}+9a^{2}+2\right)-7\left(a^{2}-4^{2}\right)+4\left(a+1\right)$$

Use the distributive property to multiply 4 by a^{2}+2a+1.

$$2a^{2}+12a+18-7\left(a+4\right)\left(a-4\right)+4a^{2}+8a+4-2\left(a^{2}+9a^{2}+2\right)-7\left(a^{2}-4^{2}\right)+4\left(a+1\right)$$

Combine a^{2} and 9a^{2} to get 10a^{2}.

$$2a^{2}+12a+18-7\left(a+4\right)\left(a-4\right)+4a^{2}+8a+4-2\left(10a^{2}+2\right)-7\left(a^{2}-4^{2}\right)+4\left(a+1\right)$$

Use the distributive property to multiply -2 by 10a^{2}+2.

$$2a^{2}+12a+18-7\left(a+4\right)\left(a-4\right)+4a^{2}+8a+4-20a^{2}-4-7\left(a^{2}-4^{2}\right)+4\left(a+1\right)$$

Combine 4a^{2} and -20a^{2} to get -16a^{2}.

$$2a^{2}+12a+18-7\left(a+4\right)\left(a-4\right)-16a^{2}+8a+4-4-7\left(a^{2}-4^{2}\right)+4\left(a+1\right)$$

Subtract 4 from 4 to get 0.

$$2a^{2}+12a+18-7\left(a+4\right)\left(a-4\right)-16a^{2}+8a-7\left(a^{2}-4^{2}\right)+4\left(a+1\right)$$

Calculate 4 to the power of 2 and get 16.

$$2a^{2}+12a+18-7\left(a+4\right)\left(a-4\right)-16a^{2}+8a-7\left(a^{2}-16\right)+4\left(a+1\right)$$

Use the distributive property to multiply -7 by a^{2}-16.

$$2a^{2}+12a+18-7\left(a+4\right)\left(a-4\right)-16a^{2}+8a-7a^{2}+112+4\left(a+1\right)$$

Combine -16a^{2} and -7a^{2} to get -23a^{2}.

$$2a^{2}+12a+18-7\left(a+4\right)\left(a-4\right)-23a^{2}+8a+112+4\left(a+1\right)$$

Use the distributive property to multiply 4 by a+1.

$$2a^{2}+12a+18-7\left(a+4\right)\left(a-4\right)-23a^{2}+8a+112+4a+4$$

Combine 8a and 4a to get 12a.

$$2a^{2}+12a+18-7\left(a+4\right)\left(a-4\right)-23a^{2}+12a+112+4$$

Add 112 and 4 to get 116.

$$2a^{2}+12a+18-7\left(a+4\right)\left(a-4\right)-23a^{2}+12a+116$$

Use the distributive property to multiply -7 by a+4.

$$2a^{2}+12a+18+\left(-7a-28\right)\left(a-4\right)-23a^{2}+12a+116$$

Use the distributive property to multiply -7a-28 by a-4 and combine like terms.

$$2a^{2}+12a+18-7a^{2}+112-23a^{2}+12a+116$$

Combine 2a^{2} and -7a^{2} to get -5a^{2}.

$$-5a^{2}+12a+18+112-23a^{2}+12a+116$$

Add 18 and 112 to get 130.

$$-5a^{2}+12a+130-23a^{2}+12a+116$$

Combine -5a^{2} and -23a^{2} to get -28a^{2}.

$$-28a^{2}+12a+130+12a+116$$

Combine 12a and 12a to get 24a.

$$-28a^{2}+24a+130+116$$

Add 130 and 116 to get 246.

$$-28a^{2}+24a+246$$

Expand

$$246+24a-28a^{2}$$

Solution Steps

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(a+3\right)^{2}.

$$2\left(a^{2}+6a+9\right)-7\left(a+4\right)\left(a-4\right)+4\left(-a-1\right)^{2}-2\left(a^{2}+9a^{2}+2\right)-7\left(a^{2}-4^{2}\right)+4\left(a+1\right)$$

Use the distributive property to multiply 2 by a^{2}+6a+9.

$$2a^{2}+12a+18-7\left(a+4\right)\left(a-4\right)+4\left(-a-1\right)^{2}-2\left(a^{2}+9a^{2}+2\right)-7\left(a^{2}-4^{2}\right)+4\left(a+1\right)$$

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(-a-1\right)^{2}.

$$2a^{2}+12a+18-7\left(a+4\right)\left(a-4\right)+4\left(\left(-a\right)^{2}-2\left(-a\right)+1\right)-2\left(a^{2}+9a^{2}+2\right)-7\left(a^{2}-4^{2}\right)+4\left(a+1\right)$$

Calculate -a to the power of 2 and get a^{2}.

$$2a^{2}+12a+18-7\left(a+4\right)\left(a-4\right)+4\left(a^{2}-2\left(-a\right)+1\right)-2\left(a^{2}+9a^{2}+2\right)-7\left(a^{2}-4^{2}\right)+4\left(a+1\right)$$

Multiply -2 and -1 to get 2.

$$2a^{2}+12a+18-7\left(a+4\right)\left(a-4\right)+4\left(a^{2}+2a+1\right)-2\left(a^{2}+9a^{2}+2\right)-7\left(a^{2}-4^{2}\right)+4\left(a+1\right)$$

Use the distributive property to multiply 4 by a^{2}+2a+1.

$$2a^{2}+12a+18-7\left(a+4\right)\left(a-4\right)+4a^{2}+8a+4-2\left(a^{2}+9a^{2}+2\right)-7\left(a^{2}-4^{2}\right)+4\left(a+1\right)$$

Combine a^{2} and 9a^{2} to get 10a^{2}.

$$2a^{2}+12a+18-7\left(a+4\right)\left(a-4\right)+4a^{2}+8a+4-2\left(10a^{2}+2\right)-7\left(a^{2}-4^{2}\right)+4\left(a+1\right)$$

Use the distributive property to multiply -2 by 10a^{2}+2.

$$2a^{2}+12a+18-7\left(a+4\right)\left(a-4\right)+4a^{2}+8a+4-20a^{2}-4-7\left(a^{2}-4^{2}\right)+4\left(a+1\right)$$

Combine 4a^{2} and -20a^{2} to get -16a^{2}.

$$2a^{2}+12a+18-7\left(a+4\right)\left(a-4\right)-16a^{2}+8a+4-4-7\left(a^{2}-4^{2}\right)+4\left(a+1\right)$$

Subtract 4 from 4 to get 0.

$$2a^{2}+12a+18-7\left(a+4\right)\left(a-4\right)-16a^{2}+8a-7\left(a^{2}-4^{2}\right)+4\left(a+1\right)$$

Calculate 4 to the power of 2 and get 16.

$$2a^{2}+12a+18-7\left(a+4\right)\left(a-4\right)-16a^{2}+8a-7\left(a^{2}-16\right)+4\left(a+1\right)$$

Use the distributive property to multiply -7 by a^{2}-16.

$$2a^{2}+12a+18-7\left(a+4\right)\left(a-4\right)-16a^{2}+8a-7a^{2}+112+4\left(a+1\right)$$

Combine -16a^{2} and -7a^{2} to get -23a^{2}.

$$2a^{2}+12a+18-7\left(a+4\right)\left(a-4\right)-23a^{2}+8a+112+4\left(a+1\right)$$

Use the distributive property to multiply 4 by a+1.

$$2a^{2}+12a+18-7\left(a+4\right)\left(a-4\right)-23a^{2}+8a+112+4a+4$$

Combine 8a and 4a to get 12a.

$$2a^{2}+12a+18-7\left(a+4\right)\left(a-4\right)-23a^{2}+12a+112+4$$

Add 112 and 4 to get 116.

$$2a^{2}+12a+18-7\left(a+4\right)\left(a-4\right)-23a^{2}+12a+116$$

Use the distributive property to multiply -7 by a+4.

$$2a^{2}+12a+18+\left(-7a-28\right)\left(a-4\right)-23a^{2}+12a+116$$

Use the distributive property to multiply -7a-28 by a-4 and combine like terms.

$$2a^{2}+12a+18-7a^{2}+112-23a^{2}+12a+116$$

Combine 2a^{2} and -7a^{2} to get -5a^{2}.

$$-5a^{2}+12a+18+112-23a^{2}+12a+116$$

Add 18 and 112 to get 130.

$$-5a^{2}+12a+130-23a^{2}+12a+116$$

Combine -5a^{2} and -23a^{2} to get -28a^{2}.

$$-28a^{2}+12a+130+12a+116$$

Combine 12a and 12a to get 24a.

$$-28a^{2}+24a+130+116$$

Add 130 and 116 to get 246.

$$-28a^{2}+24a+246$$