Solve [(2b-3a)*(2a+b)-(a+2b)^2]+1/2a+1/2b-(5+2/3a)=

$$[ ( 2 b - 3 a ) \cdot ( 2 a + b ) - ( a + 2 b ) ^ { 2 } ] + \frac { 1 } { 2 } a + \frac { 1 } { 2 } b - ( 5 + \frac { 2 } { 3 } a ) =$$

Evaluate

$$-3ab-2b^{2}-7a^{2}+\frac{b}{2}-\frac{a}{6}-5$$

Solution Steps

Use the distributive property to multiply 2b-3a by 2a+b and combine like terms.

$$ba+2b^{2}-6a^{2}-\left(a+2b\right)^{2}+\frac{1}{2}a+\frac{1}{2}b-\left(5+\frac{2}{3}a\right)$$

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

$$ba+2b^{2}-6a^{2}-\left(a^{2}+4ab+4b^{2}\right)+\frac{1}{2}a+\frac{1}{2}b-\left(5+\frac{2}{3}a\right)$$

To find the opposite of a^{2}+4ab+4b^{2}, find the opposite of each term.

$$ba+2b^{2}-6a^{2}-a^{2}-4ab-4b^{2}+\frac{1}{2}a+\frac{1}{2}b-\left(5+\frac{2}{3}a\right)$$

Combine -6a^{2} and -a^{2} to get -7a^{2}.

$$ba+2b^{2}-7a^{2}-4ab-4b^{2}+\frac{1}{2}a+\frac{1}{2}b-\left(5+\frac{2}{3}a\right)$$

Combine ba and -4ab to get -3ba.

$$-3ba+2b^{2}-7a^{2}-4b^{2}+\frac{1}{2}a+\frac{1}{2}b-\left(5+\frac{2}{3}a\right)$$

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

$$-3ba-2b^{2}-7a^{2}+\frac{1}{2}a+\frac{1}{2}b-\left(5+\frac{2}{3}a\right)$$

To find the opposite of 5+\frac{2}{3}a, find the opposite of each term.

$$-3ba-2b^{2}-7a^{2}+\frac{1}{2}a+\frac{1}{2}b-5-\frac{2}{3}a$$

Combine \frac{1}{2}a and -\frac{2}{3}a to get -\frac{1}{6}a.

$$-3ba-2b^{2}-7a^{2}-\frac{1}{6}a+\frac{1}{2}b-5$$

Expand

$$-3ab-2b^{2}-7a^{2}+\frac{b}{2}-\frac{a}{6}-5$$

Solution Steps

Use the distributive property to multiply 2b-3a by 2a+b and combine like terms.

$$ba+2b^{2}-6a^{2}-\left(a+2b\right)^{2}+\frac{1}{2}a+\frac{1}{2}b-\left(5+\frac{2}{3}a\right)$$

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

$$ba+2b^{2}-6a^{2}-\left(a^{2}+4ab+4b^{2}\right)+\frac{1}{2}a+\frac{1}{2}b-\left(5+\frac{2}{3}a\right)$$

To find the opposite of a^{2}+4ab+4b^{2}, find the opposite of each term.

$$ba+2b^{2}-6a^{2}-a^{2}-4ab-4b^{2}+\frac{1}{2}a+\frac{1}{2}b-\left(5+\frac{2}{3}a\right)$$

Combine -6a^{2} and -a^{2} to get -7a^{2}.

$$ba+2b^{2}-7a^{2}-4ab-4b^{2}+\frac{1}{2}a+\frac{1}{2}b-\left(5+\frac{2}{3}a\right)$$

Combine ba and -4ab to get -3ba.

$$-3ba+2b^{2}-7a^{2}-4b^{2}+\frac{1}{2}a+\frac{1}{2}b-\left(5+\frac{2}{3}a\right)$$

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

$$-3ba-2b^{2}-7a^{2}+\frac{1}{2}a+\frac{1}{2}b-\left(5+\frac{2}{3}a\right)$$

To find the opposite of 5+\frac{2}{3}a, find the opposite of each term.

$$-3ba-2b^{2}-7a^{2}+\frac{1}{2}a+\frac{1}{2}b-5-\frac{2}{3}a$$

Combine \frac{1}{2}a and -\frac{2}{3}a to get -\frac{1}{6}a.

$$-3ba-2b^{2}-7a^{2}-\frac{1}{6}a+\frac{1}{2}b-5$$