Question

d) $(-6 + 2m)(-6 + 2m)$
e) $(-8 - 2x)(3 - 7x)$
f) $(6 - 5n)(-6 + 5n)$

Explanation:

To solve these multiplication problems, we'll use the distributive property (also known as the FOIL method for binomials) where \((a + b)(c + d)=ac + ad+bc + bd\), or for the difference of squares \((a - b)(a + b)=a^2 - b^2\) (if applicable). Let's solve each part:

Part (d): \((-6 + 2m)(-6 + 2m)\)

This is a square of a binomial \((a + b)^2=a^2 + 2ab + b^2\), where \(a=-6\) and \(b = 2m\).

Step 1: Identify \(a\) and \(b\)

Here, \(a=-6\) and \(b = 2m\).

Step 2: Apply the formula \((a + b)^2=a^2+2ab + b^2\)

• Calculate \(a^2\): \((-6)^2=36\)
• Calculate \(2ab\): \(2\times(-6)\times(2m)=-24m\)
• Calculate \(b^2\): \((2m)^2 = 4m^2\)

Step 3: Combine the terms

\((-6 + 2m)^2=36-24m + 4m^2\) (we can also write it as \(4m^2-24m + 36\))

Part (e): \((-8 - 2x)(3 - 7x)\)

We'll use the distributive property (FOIL method). Let \(a=-8\), \(b=-2x\), \(c = 3\), \(d=-7x\).

Step 1: Multiply the First terms: \(a\times c\)

\((-8)\times(3)=-24\)

Step 2: Multiply the Outer terms: \(a\times d\)

\((-8)\times(-7x)=56x\)

Step 3: Multiply the Inner terms: \(b\times c\)

\((-2x)\times(3)=-6x\)

Step 4: Multiply the Last terms: \(b\times d\)

\((-2x)\times(-7x)=14x^2\)

Step 5: Combine like terms

First, combine the \(x\)-terms: \(56x-6x = 50x\)
Then, combine all terms: \(-24 + 50x+14x^2\) (or \(14x^2+50x - 24\))

Part (f): \((6 - 5n)(-6 + 5n)\)

Notice that this is of the form \((a - b)(-a + b)=-(a - b)(a - b)=-(a - b)^2\) (since \(-a + b=-(a - b)\)). Let \(a = 6\) and \(b = 5n\).

Step 1: Rewrite the expression

\((6 - 5n)(-6 + 5n)=(6 - 5n)(-(6 - 5n))=-(6 - 5n)^2\)

Step 2: Apply the formula \((a - b)^2=a^2-2ab + b^2\) (where \(a = 6\) and \(b = 5n\))

• Calculate \(a^2\): \(6^2 = 36\)
• Calculate \(2ab\): \(2\times6\times(5n)=60n\)
• Calculate \(b^2\): \((5n)^2=25n^2\)

Step 3: Substitute into \((a - b)^2\)

\((6 - 5n)^2=36-60n + 25n^2\)

Step 4: Apply the negative sign

\(-(6 - 5n)^2=-36 + 60n-25n^2\) (or \(-25n^2+60n - 36\))

Final Answers:

• (d): \(\boldsymbol{4m^2-24m + 36}\) (or \(36-24m + 4m^2\))
• (e): \(\boldsymbol{14x^2+50x - 24}\) (or \(-24 + 50x+14x^2\))
• (f): \(\boldsymbol{-25n^2+60n - 36}\) (or \(-36 + 60n-25n^2\))

Answer:

To solve these multiplication problems, we'll use the distributive property (also known as the FOIL method for binomials) where \((a + b)(c + d)=ac + ad+bc + bd\), or for the difference of squares \((a - b)(a + b)=a^2 - b^2\) (if applicable). Let's solve each part:

Part (d): \((-6 + 2m)(-6 + 2m)\)

This is a square of a binomial \((a + b)^2=a^2 + 2ab + b^2\), where \(a=-6\) and \(b = 2m\).

Step 1: Identify \(a\) and \(b\)

Here, \(a=-6\) and \(b = 2m\).

Step 2: Apply the formula \((a + b)^2=a^2+2ab + b^2\)

• Calculate \(a^2\): \((-6)^2=36\)
• Calculate \(2ab\): \(2\times(-6)\times(2m)=-24m\)
• Calculate \(b^2\): \((2m)^2 = 4m^2\)

Step 3: Combine the terms

\((-6 + 2m)^2=36-24m + 4m^2\) (we can also write it as \(4m^2-24m + 36\))

Part (e): \((-8 - 2x)(3 - 7x)\)

We'll use the distributive property (FOIL method). Let \(a=-8\), \(b=-2x\), \(c = 3\), \(d=-7x\).

Step 1: Multiply the First terms: \(a\times c\)

\((-8)\times(3)=-24\)

Step 2: Multiply the Outer terms: \(a\times d\)

\((-8)\times(-7x)=56x\)

Step 3: Multiply the Inner terms: \(b\times c\)

\((-2x)\times(3)=-6x\)

Step 4: Multiply the Last terms: \(b\times d\)

\((-2x)\times(-7x)=14x^2\)

Step 5: Combine like terms

First, combine the \(x\)-terms: \(56x-6x = 50x\)
Then, combine all terms: \(-24 + 50x+14x^2\) (or \(14x^2+50x - 24\))

Part (f): \((6 - 5n)(-6 + 5n)\)

Notice that this is of the form \((a - b)(-a + b)=-(a - b)(a - b)=-(a - b)^2\) (since \(-a + b=-(a - b)\)). Let \(a = 6\) and \(b = 5n\).

Step 1: Rewrite the expression

\((6 - 5n)(-6 + 5n)=(6 - 5n)(-(6 - 5n))=-(6 - 5n)^2\)

Step 2: Apply the formula \((a - b)^2=a^2-2ab + b^2\) (where \(a = 6\) and \(b = 5n\))

• Calculate \(a^2\): \(6^2 = 36\)
• Calculate \(2ab\): \(2\times6\times(5n)=60n\)
• Calculate \(b^2\): \((5n)^2=25n^2\)

Step 3: Substitute into \((a - b)^2\)

\((6 - 5n)^2=36-60n + 25n^2\)

Step 4: Apply the negative sign

\(-(6 - 5n)^2=-36 + 60n-25n^2\) (or \(-25n^2+60n - 36\))

Final Answers:

• (d): \(\boldsymbol{4m^2-24m + 36}\) (or \(36-24m + 4m^2\))
• (e): \(\boldsymbol{14x^2+50x - 24}\) (or \(-24 + 50x+14x^2\))
• (f): \(\boldsymbol{-25n^2+60n - 36}\) (or \(-36 + 60n-25n^2\))