Answer :
Let's factor each of the given expressions step-by-step:
5) [tex]\( m^4 + 8m^2 + 12 \)[/tex]
1. Introduce a substitution: Let [tex]\( u = m^2 \)[/tex]. Then the equation becomes [tex]\( u^2 + 8u + 12 \)[/tex].
2. Factor the quadratic [tex]\( u^2 + 8u + 12 \)[/tex] into [tex]\( (u + 2)(u + 6) \)[/tex].
3. Substitute back [tex]\( u = m^2 \)[/tex] to get: [tex]\((m^2 + 2)(m^2 + 6)\)[/tex].
The factored form is [tex]\((m^2 + 2)(m^2 + 6)\)[/tex].
6) [tex]\( x^4 - 2x^2 - 15 \)[/tex]
1. Introduce a substitution: Let [tex]\( v = x^2 \)[/tex]. Then the equation becomes [tex]\( v^2 - 2v - 15 \)[/tex].
2. Factor the quadratic [tex]\( v^2 - 2v - 15 \)[/tex] into [tex]\( (v - 5)(v + 3) \)[/tex].
3. Substitute back [tex]\( v = x^2 \)[/tex] to get: [tex]\((x^2 - 5)(x^2 + 3)\)[/tex].
The factored form is [tex]\((x^2 - 5)(x^2 + 3)\)[/tex].
7) [tex]\( x^4 - 8x^2 + 7 \)[/tex]
1. Introduce a substitution: Let [tex]\( w = x^2 \)[/tex]. Then the equation becomes [tex]\( w^2 - 8w + 7 \)[/tex].
2. Factor the quadratic [tex]\( w^2 - 8w + 7 \)[/tex] into [tex]\( (w - 7)(w - 1) \)[/tex].
3. Substitute back [tex]\( w = x^2 \)[/tex] to get: [tex]\((x^2 - 7)(x^2 - 1)\)[/tex].
The factored form is [tex]\((x^2 - 7)(x^2 - 1)\)[/tex].
8) [tex]\( 2x^4 + 2x^2 - 144 \)[/tex]
1. Introduce a substitution: Let [tex]\( y = x^2 \)[/tex]. Then the equation becomes [tex]\( 2y^2 + 2y - 144 \)[/tex].
2. Factor the quadratic [tex]\( 2y^2 + 2y - 144 \)[/tex] to get: [tex]\( 2(y^2 + y - 72) \)[/tex].
3. Factor [tex]\( y^2 + y - 72 \)[/tex] into [tex]\( (y - 8)(y + 9) \)[/tex].
4. Substitute back [tex]\( y = x^2 \)[/tex] to get: [tex]\( 2(x^2 - 8)(x^2 + 9)\)[/tex].
The factored form is [tex]\( 2(x^2 - 8)(x^2 + 9) \)[/tex].
9) [tex]\( 3x^6 - 24x^4 + 45x^2 \)[/tex]
1. Factor out the common term [tex]\( 3x^2 \)[/tex]: [tex]\( 3x^2(x^4 - 8x^2 + 15) \)[/tex].
2. Introduce a substitution: Let [tex]\( z = x^2 \)[/tex]. Then the quadratic becomes [tex]\( z^2 - 8z + 15 \)[/tex].
3. Factor [tex]\( z^2 - 8z + 15 \)[/tex] into [tex]\( (z - 5)(z - 3) \)[/tex].
4. Substitute back [tex]\( z = x^2 \)[/tex] to get: [tex]\( 3x^2(x^2 - 5)(x^2 - 3)\)[/tex].
The factored form is [tex]\( 3x^2(x^2 - 5)(x^2 - 3) \)[/tex].
10) [tex]\( x^5 - 2x^3 - 63x \)[/tex]
1. Factor out the common term [tex]\( x \)[/tex]: [tex]\( x(x^4 - 2x^2 - 63) \)[/tex].
2. Introduce a substitution: Let [tex]\( a = x^2 \)[/tex]. Then the quadratic becomes [tex]\( a^2 - 2a - 63 \)[/tex].
3. Factor [tex]\( a^2 - 2a - 63 \)[/tex] into [tex]\( (a - 9)(a + 7) \)[/tex].
4. Substitute back [tex]\( a = x^2 \)[/tex] to get: [tex]\( x((x^2 - 9)(x^2 + 7)) \)[/tex].
5. Further factor [tex]\( x^2 - 9 \)[/tex] as [tex]\((x - 3)(x + 3)\)[/tex].
The factored form is [tex]\( x(x - 3)(x + 3)(x^2 + 7) \)[/tex].
These are the detailed factored forms for each expression.
5) [tex]\( m^4 + 8m^2 + 12 \)[/tex]
1. Introduce a substitution: Let [tex]\( u = m^2 \)[/tex]. Then the equation becomes [tex]\( u^2 + 8u + 12 \)[/tex].
2. Factor the quadratic [tex]\( u^2 + 8u + 12 \)[/tex] into [tex]\( (u + 2)(u + 6) \)[/tex].
3. Substitute back [tex]\( u = m^2 \)[/tex] to get: [tex]\((m^2 + 2)(m^2 + 6)\)[/tex].
The factored form is [tex]\((m^2 + 2)(m^2 + 6)\)[/tex].
6) [tex]\( x^4 - 2x^2 - 15 \)[/tex]
1. Introduce a substitution: Let [tex]\( v = x^2 \)[/tex]. Then the equation becomes [tex]\( v^2 - 2v - 15 \)[/tex].
2. Factor the quadratic [tex]\( v^2 - 2v - 15 \)[/tex] into [tex]\( (v - 5)(v + 3) \)[/tex].
3. Substitute back [tex]\( v = x^2 \)[/tex] to get: [tex]\((x^2 - 5)(x^2 + 3)\)[/tex].
The factored form is [tex]\((x^2 - 5)(x^2 + 3)\)[/tex].
7) [tex]\( x^4 - 8x^2 + 7 \)[/tex]
1. Introduce a substitution: Let [tex]\( w = x^2 \)[/tex]. Then the equation becomes [tex]\( w^2 - 8w + 7 \)[/tex].
2. Factor the quadratic [tex]\( w^2 - 8w + 7 \)[/tex] into [tex]\( (w - 7)(w - 1) \)[/tex].
3. Substitute back [tex]\( w = x^2 \)[/tex] to get: [tex]\((x^2 - 7)(x^2 - 1)\)[/tex].
The factored form is [tex]\((x^2 - 7)(x^2 - 1)\)[/tex].
8) [tex]\( 2x^4 + 2x^2 - 144 \)[/tex]
1. Introduce a substitution: Let [tex]\( y = x^2 \)[/tex]. Then the equation becomes [tex]\( 2y^2 + 2y - 144 \)[/tex].
2. Factor the quadratic [tex]\( 2y^2 + 2y - 144 \)[/tex] to get: [tex]\( 2(y^2 + y - 72) \)[/tex].
3. Factor [tex]\( y^2 + y - 72 \)[/tex] into [tex]\( (y - 8)(y + 9) \)[/tex].
4. Substitute back [tex]\( y = x^2 \)[/tex] to get: [tex]\( 2(x^2 - 8)(x^2 + 9)\)[/tex].
The factored form is [tex]\( 2(x^2 - 8)(x^2 + 9) \)[/tex].
9) [tex]\( 3x^6 - 24x^4 + 45x^2 \)[/tex]
1. Factor out the common term [tex]\( 3x^2 \)[/tex]: [tex]\( 3x^2(x^4 - 8x^2 + 15) \)[/tex].
2. Introduce a substitution: Let [tex]\( z = x^2 \)[/tex]. Then the quadratic becomes [tex]\( z^2 - 8z + 15 \)[/tex].
3. Factor [tex]\( z^2 - 8z + 15 \)[/tex] into [tex]\( (z - 5)(z - 3) \)[/tex].
4. Substitute back [tex]\( z = x^2 \)[/tex] to get: [tex]\( 3x^2(x^2 - 5)(x^2 - 3)\)[/tex].
The factored form is [tex]\( 3x^2(x^2 - 5)(x^2 - 3) \)[/tex].
10) [tex]\( x^5 - 2x^3 - 63x \)[/tex]
1. Factor out the common term [tex]\( x \)[/tex]: [tex]\( x(x^4 - 2x^2 - 63) \)[/tex].
2. Introduce a substitution: Let [tex]\( a = x^2 \)[/tex]. Then the quadratic becomes [tex]\( a^2 - 2a - 63 \)[/tex].
3. Factor [tex]\( a^2 - 2a - 63 \)[/tex] into [tex]\( (a - 9)(a + 7) \)[/tex].
4. Substitute back [tex]\( a = x^2 \)[/tex] to get: [tex]\( x((x^2 - 9)(x^2 + 7)) \)[/tex].
5. Further factor [tex]\( x^2 - 9 \)[/tex] as [tex]\((x - 3)(x + 3)\)[/tex].
The factored form is [tex]\( x(x - 3)(x + 3)(x^2 + 7) \)[/tex].
These are the detailed factored forms for each expression.
