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

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

Evaluate

$$17a+61$$

Solution Steps

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

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

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

$$4a^{2}+16a+16-7\left(a+2\right)\left(a-3\right)+3\left(a-1\right)^{2}$$

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

$$4a^{2}+16a+16-7\left(a+2\right)\left(a-3\right)+3\left(a^{2}-2a+1\right)$$

Use the distributive property to multiply 3 by a^{2}-2a+1.

$$4a^{2}+16a+16-7\left(a+2\right)\left(a-3\right)+3a^{2}-6a+3$$

Use the distributive property to multiply -7 by a+2.

$$4a^{2}+16a+16+\left(-7a-14\right)\left(a-3\right)+3a^{2}-6a+3$$

Use the distributive property to multiply -7a-14 by a-3 and combine like terms.

$$4a^{2}+16a+16-7a^{2}+7a+42+3a^{2}-6a+3$$

Combine 4a^{2} and -7a^{2} to get -3a^{2}.

$$-3a^{2}+16a+16+7a+42+3a^{2}-6a+3$$

Combine 16a and 7a to get 23a.

$$-3a^{2}+23a+16+42+3a^{2}-6a+3$$

Add 16 and 42 to get 58.

$$-3a^{2}+23a+58+3a^{2}-6a+3$$

Combine -3a^{2} and 3a^{2} to get 0.

$$23a+58-6a+3$$

Combine 23a and -6a to get 17a.

$$17a+58+3$$

Add 58 and 3 to get 61.

$$17a+61$$

Expand

$$17a+61$$

Solution Steps

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

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

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

$$4a^{2}+16a+16-7\left(a+2\right)\left(a-3\right)+3\left(a-1\right)^{2}$$

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

$$4a^{2}+16a+16-7\left(a+2\right)\left(a-3\right)+3\left(a^{2}-2a+1\right)$$

Use the distributive property to multiply 3 by a^{2}-2a+1.

$$4a^{2}+16a+16-7\left(a+2\right)\left(a-3\right)+3a^{2}-6a+3$$

Use the distributive property to multiply -7 by a+2.

$$4a^{2}+16a+16+\left(-7a-14\right)\left(a-3\right)+3a^{2}-6a+3$$

Use the distributive property to multiply -7a-14 by a-3 and combine like terms.

$$4a^{2}+16a+16-7a^{2}+7a+42+3a^{2}-6a+3$$

Combine 4a^{2} and -7a^{2} to get -3a^{2}.

$$-3a^{2}+16a+16+7a+42+3a^{2}-6a+3$$

Combine 16a and 7a to get 23a.

$$-3a^{2}+23a+16+42+3a^{2}-6a+3$$

Add 16 and 42 to get 58.

$$-3a^{2}+23a+58+3a^{2}-6a+3$$

Combine -3a^{2} and 3a^{2} to get 0.

$$23a+58-6a+3$$

Combine 23a and -6a to get 17a.

$$17a+58+3$$

Add 58 and 3 to get 61.

$$17a+61$$