Solve (a-2)^3+(a-5)(a+5)-a(1+a^2)=-5a^2+4a-2

$$( a - 2 ) ^ { 3 } + ( a - 5 ) ( a + 5 ) - a ( 1 + a ^ { 2 } ) = - 5 a ^ { 2 } + 4 a - 2$$

Solve for a

$$a = \frac{31}{7} = 4\frac{3}{7} \approx 4.428571429$$

Steps for Solving Linear Equation

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(a-2\right)^{3}.

$$a^{3}-6a^{2}+12a-8+\left(a-5\right)\left(a+5\right)-a\left(1+a^{2}\right)=-5a^{2}+4a-2$$

Consider \left(a-5\right)\left(a+5\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 5.

$$a^{3}-6a^{2}+12a-8+a^{2}-25-a\left(1+a^{2}\right)=-5a^{2}+4a-2$$

Combine -6a^{2} and a^{2} to get -5a^{2}.

$$a^{3}-5a^{2}+12a-8-25-a\left(1+a^{2}\right)=-5a^{2}+4a-2$$

Subtract 25 from -8 to get -33.

$$a^{3}-5a^{2}+12a-33-a\left(1+a^{2}\right)=-5a^{2}+4a-2$$

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

$$a^{3}-5a^{2}+12a-33-\left(a+a^{3}\right)=-5a^{2}+4a-2$$

To find the opposite of a+a^{3}, find the opposite of each term.

$$a^{3}-5a^{2}+12a-33-a-a^{3}=-5a^{2}+4a-2$$

Combine 12a and -a to get 11a.

$$a^{3}-5a^{2}+11a-33-a^{3}=-5a^{2}+4a-2$$

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

$$-5a^{2}+11a-33=-5a^{2}+4a-2$$

Add 5a^{2} to both sides.

$$-5a^{2}+11a-33+5a^{2}=4a-2$$

Combine -5a^{2} and 5a^{2} to get 0.

$$11a-33=4a-2$$

Subtract 4a from both sides.

$$11a-33-4a=-2$$

Combine 11a and -4a to get 7a.

$$7a-33=-2$$

Add 33 to both sides.

$$7a=-2+33$$

Add -2 and 33 to get 31.

$$7a=31$$

Divide both sides by 7.

$$a=\frac{31}{7}$$