Solve (a-a^2-1)^2-3(a+2)^2=a^3(a-1)+a[(a+2)(1-a)+a]

$$( a - a ^ { 2 } - 1 ) ^ { 2 } - 3 ( a + 2 ) ^ { 2 } = a ^ { 3 } ( a - 1 ) + a [ ( a + 2 ) ( 1 - a ) + a ]$$

Solve for a

$$a=-\frac{11}{16}=-0.6875$$

Steps for Solving Linear Equation

Square a-a^{2}-1.

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

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

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

Use the distributive property to multiply -3 by a^{2}+4a+4.

$$a^{4}-2a^{3}+3a^{2}-2a+1-3a^{2}-12a-12=a^{3}\left(a-1\right)+a\left(\left(a+2\right)\left(1-a\right)+a\right)$$

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

$$a^{4}-2a^{3}-2a+1-12a-12=a^{3}\left(a-1\right)+a\left(\left(a+2\right)\left(1-a\right)+a\right)$$

Combine -2a and -12a to get -14a.

$$a^{4}-2a^{3}-14a+1-12=a^{3}\left(a-1\right)+a\left(\left(a+2\right)\left(1-a\right)+a\right)$$

Subtract 12 from 1 to get -11.

$$a^{4}-2a^{3}-14a-11=a^{3}\left(a-1\right)+a\left(\left(a+2\right)\left(1-a\right)+a\right)$$

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

$$a^{4}-2a^{3}-14a-11=a^{4}-a^{3}+a\left(\left(a+2\right)\left(1-a\right)+a\right)$$

Use the distributive property to multiply a+2 by 1-a and combine like terms.

$$a^{4}-2a^{3}-14a-11=a^{4}-a^{3}+a\left(-a-a^{2}+2+a\right)$$

Combine -a and a to get 0.

$$a^{4}-2a^{3}-14a-11=a^{4}-a^{3}+a\left(-a^{2}+2\right)$$

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

$$a^{4}-2a^{3}-14a-11=a^{4}-a^{3}-a^{3}+2a$$

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

$$a^{4}-2a^{3}-14a-11=a^{4}-2a^{3}+2a$$

Subtract a^{4} from both sides.

$$a^{4}-2a^{3}-14a-11-a^{4}=-2a^{3}+2a$$

Combine a^{4} and -a^{4} to get 0.

$$-2a^{3}-14a-11=-2a^{3}+2a$$

Add 2a^{3} to both sides.

$$-2a^{3}-14a-11+2a^{3}=2a$$

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

$$-14a-11=2a$$

Subtract 2a from both sides.

$$-14a-11-2a=0$$

Combine -14a and -2a to get -16a.

$$-16a-11=0$$

Add 11 to both sides. Anything plus zero gives itself.

$$-16a=11$$

Divide both sides by -16.

$$a=\frac{11}{-16}$$

Fraction \frac{11}{-16} can be rewritten as -\frac{11}{16} by extracting the negative sign.

$$a=-\frac{11}{16}$$