Question

find the missing term.
1
$(4x + 3) (9x^2 + 7x - 3) = 36x^3 + 55x^2 + ? - 9$
$? = $
2
$(2x^2 + 7x - 6) (5x + 9) = ? +53x^2 + 33x - 54$
$? = $
find the area of the shape.
triangle with height $2x^2$ and base $3x^2 + 5x - 6$

Explanation:

Problem 1:

Step1: Expand the left - hand side

We use the distributive property (also known as the FOIL method for binomials, but extended for polynomials) to expand \((4x + 3)(9x^{2}+7x - 3)\).
First, multiply \(4x\) by each term in \((9x^{2}+7x - 3)\):
\(4x\times9x^{2}=36x^{3}\), \(4x\times7x = 28x^{2}\), \(4x\times(-3)=-12x\)
Then, multiply \(3\) by each term in \((9x^{2}+7x - 3)\):
\(3\times9x^{2}=27x^{2}\), \(3\times7x = 21x\), \(3\times(-3)=-9\)

Step2: Combine like terms

Now, combine the terms we got from the expansion:
For the \(x^{3}\) term: \(36x^{3}\)
For the \(x^{2}\) term: \(28x^{2}+27x^{2}=55x^{2}\)
For the \(x\) term: \(-12x + 21x=9x\)
For the constant term: \(-9\)

So, \((4x + 3)(9x^{2}+7x - 3)=36x^{3}+55x^{2}+9x - 9\)

Step1: Expand the left - hand side

Use the distributive property to expand \((2x^{2}+7x - 6)(5x + 9)\).
Multiply \(2x^{2}\) by each term in \((5x + 9)\): \(2x^{2}\times5x = 10x^{3}\), \(2x^{2}\times9=18x^{2}\)
Multiply \(7x\) by each term in \((5x + 9)\): \(7x\times5x = 35x^{2}\), \(7x\times9 = 63x\)
Multiply \(-6\) by each term in \((5x + 9)\): \(-6\times5x=-30x\), \(-6\times9=-54\)

Step2: Combine like terms

Combine the terms:
For the \(x^{3}\) term: \(10x^{3}\)
For the \(x^{2}\) term: \(18x^{2}+35x^{2}=53x^{2}\)
For the \(x\) term: \(63x-30x = 33x\)
For the constant term: \(-54\)

So, \((2x^{2}+7x - 6)(5x + 9)=10x^{3}+53x^{2}+33x - 54\)

Step1: Apply the area formula

\(A=\frac{1}{2}\times(3x^{2}+5x - 6)\times(2x^{2})\)

Step2: Simplify the expression

We can rewrite the expression as \((3x^{2}+5x - 6)\times x^{2}\) (because \(\frac{1}{2}\times2x^{2}=x^{2}\))
Now, use the distributive property:
\(3x^{2}\times x^{2}=3x^{4}\), \(5x\times x^{2}=5x^{3}\), \(-6\times x^{2}=-6x^{2}\)

So, the area \(A = 3x^{4}+5x^{3}-6x^{2}\)

Answer:

\(9x\)

Problem 2: