Solve 2left(a+3right)^2-7left(a+4right)left(a-4right)+4left(-a-1right)^2

$$2 { \left(a+3 \right) }^{ 2 } -7 \left( a+4 \right) \left( a-4 \right) +4 { \left(-a-1 \right) }^{ 2 }$$

Evaluate

$$134+20a-a^{2}$$

Solution Steps

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

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

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

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

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

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

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

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

Multiply -2 and -1 to get 2.

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

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

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

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

$$2a^{2}+12a+18+\left(-7a-28\right)\left(a-4\right)+4a^{2}+8a+4$$

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

$$2a^{2}+12a+18-7a^{2}+112+4a^{2}+8a+4$$

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

$$-5a^{2}+12a+18+112+4a^{2}+8a+4$$

Add 18 and 112 to get 130.

$$-5a^{2}+12a+130+4a^{2}+8a+4$$

Combine -5a^{2} and 4a^{2} to get -a^{2}.

$$-a^{2}+12a+130+8a+4$$

Combine 12a and 8a to get 20a.

$$-a^{2}+20a+130+4$$

Add 130 and 4 to get 134.

$$-a^{2}+20a+134$$

Expand

$$134+20a-a^{2}$$

Solution Steps

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

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

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

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

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

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

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

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

Multiply -2 and -1 to get 2.

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

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

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

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

$$2a^{2}+12a+18+\left(-7a-28\right)\left(a-4\right)+4a^{2}+8a+4$$

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

$$2a^{2}+12a+18-7a^{2}+112+4a^{2}+8a+4$$

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

$$-5a^{2}+12a+18+112+4a^{2}+8a+4$$

Add 18 and 112 to get 130.

$$-5a^{2}+12a+130+4a^{2}+8a+4$$

Combine -5a^{2} and 4a^{2} to get -a^{2}.

$$-a^{2}+12a+130+8a+4$$

Combine 12a and 8a to get 20a.

$$-a^{2}+20a+130+4$$

Add 130 and 4 to get 134.

$$-a^{2}+20a+134$$